Algebras, Graphs and Their Applications

ALGEBRAS, GRAPHS AND THEIR APPLICATIONS

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Algebras, Graphs and their Applications

Ilwoo ChoDepartment of Mathmatics

St. Ambros UniversityDavenport, Iowa, USA

Edited byPalle E.T. Jorgensen

University of Iowa, Iowa City, USA

$6&,(1&(38%/,6+(56%22.R

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The author really appreciates the support of his wifeOkhwa Hong

and his bright, energetic, kind, beautiful two daughtersJeanelle and Alika.

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Preface

This book aims to introduce the study of algebras induced by combinatorial objects, directed graphs. They serve as tools in an analysis of graph-theoretic problems Also conversely, some analytic problems can characterized and resolved.

These algebraic tools include groupoids, hence a subfamily called graph groupoids. Their properties are outlined and studied systematically in separate chapters. Some general ideas from the book are as follows: An action of a graph grouponid yields a dynamical system, and its study is an extension of related geometric structures, such as free-group actions.

This study is of interest in its own right, and in addition we include a host of applications: representation theory, automata theory, operator algebra (von Neumann algebra theory. C*-algebra theory, and K-theory), free probability, index theory, noncommutative dynamical systems (geometric actions of groupoids). operator theory, spectral theory, fractal analysis, information/entropy theory, and network theory.

Our systematic study of these graph groupoids is self-contained. We begin by showing that the algebraic operations of groupoid sums, product groupoids, quotient groupoids induce corresponding operations on graphs. So, on graph groupoids, the groupoidal operations are determined by the graph operations.

Since all graph groupoids act like free groups, it is natural to consider their free algebraic structures. Our starting point for this is certain operator algebras. They in turn are induced by graph groupoids. Our presentation of this includes a structure theorem based on freeness. We extend this algebraic freeness to a powerful concept of operator-algebraic freeness. Applications include a characterization of operator algebras that are induced by graph groupoids, and their representations.

We show that graph groupoid algebras are determined by free probability, making thus a link to an active area within operator algebra theory.

We study these operator algebra as Toeplitz-like algebras.Spectral information for graph groupoids is typically not easy to come by,

and our algebraic approach offers insight. For the systematic research of spectral theory in the context of graphs, we introduce the key notions of self-adjointness, unitarity, normality and hyponormality. These tools have uses in the study of classical Toeplitz operators, and a host of applied problems. It is noteworthy that the results from operator algebra and noncommutative dynamical systems lend

themselves to problems from discrete analysis; as for example in the study of distortions of histories.

By passing to limits of graphs we arrive at fractals. Here, our algebraic tools allow us to identity fractal property (or fractality) and automata theory. In this setting, one computes entropy of certain schemes. We study entropy as a measure of chaos. In particular, we compute the entropy of fractal graphs. They are graphs having their graph groupoids having fractality, called graph fractaloids.

For more advanced applications, we present index computations in von Neumann algebras. It includes the Watatanis extended Jones index. Our study not only provides good examples of Jones index computations but also offers new connections between graph inclusions and corresponding operator-algebraic inclusions.

More combinatorial applications include embeddable in networks: electric resistance networks and actions of graph groupoids.

As another application, we discuss briefly about the K0-group computations of C*-algebras generated by graph groupoids.

The author especially thanks his teacher and co-worker Dr Palle E.T. Jorgensen (Professor. University of Iowa. USA). It is a great honor for the author to work with Dr Jorgensen. Some parts of this book are based on Joint work with Dr Jorgensen.

Ilwoo ChoDepartment of Mathmatics

St. Ambros UniversityDavenport, Iowa

USA

viii Preface

Contents

Preface vii

1. Algebra on Graphs 1 1.1 Introduction, 1 1.2 Preliminaries, 2 1.2.1 Graph Groupoids, 2 1.2.2 Groupoids and Graphs, 4 1.2.3 More about Graph Groupoids, 7 1.3 Groupoids under Operations, 10 1.3.1 Sum of Groupoids, 10 1.3.2 Product Groupoids, 10 1.3.3 Quotient Groupoids, 11 1.3.4 Complemented Groupoids, 15 1.4 Operations on Graphs, 15 1.4.1 Unioned Graphs, 16 1.4.2 Product Graphs, 22 1.4.3 Quotient Graphs, 34 1.4.4 Complemented Graphs, 39 1.5 Bibliography, 41

2. Representations and Operator Algebras of Graph Groupoids 44 2.1 Introduction, 44 2.2 Partial Isometries, 45 2.3 Graph von Neumann Algebras, 46 2.3.1 Canonical Representation of Graph Groupoids, 46 2.3.2 Groupoid W*-Dynamical Systems, 48 2.3.3 Groupoid Crossed Product W*-Algebras, 50 2.4 M-Diagonal Graph W*-Probability Spaces, 52 2.4.1 Free Probability, 52 2.4.2 Free Probabilistic Models on G, 54 2.4.3 Free Structures, 55 2.4.4 Graph-Groupoid von Neumann Algebras, 76 2.4.5 Examples, 77

2.5 C*-Subalgebras Generated by Partial Isometries, 83 2.5.1 Partial Isometries and Isometric Extensions, 84 2.5.2 Directed Graphs Induced by Partial Isometries, 89 2.5.3 Groupoids Induced by Partial Isometries, 95 2.5.4 C*-Subalgbras Generated by one Partial Isometry, 97 2.5.5 Classification for C*(a), 99 2.5.6 C*-Subalgebras Generated by Partial Isometries, 103 2.5.7 Examples, 111 2.6 C*-Algebras Generated by a Single Operator, 116 2.6.1 Groupoid Crossed Product C*-Algebras, 117 2.6.2 A C*-Subalgebra of B(H) Generated by an Operator, 121 2.6.3 Examples, 126 2.7 Bibliography, 129

3. Operator Theory on Graphs 133 3.1 Introduction, 133 3.1.1 Overview, 133 3.1.2 Motivation and Applications, 135 3.2 Self-Adjointness and Unitary Property, 136 3.2.1 Graph Operators, 136 3.2.2 Self-Adjoint Graph Operators, 139 3.2.3 Unitary Graph Operators, 142 3.3 Normality of Graph Operators, 150 3.3.1 Hyponormality, 151 3.3.2 Normality, 162 3.4 Operators in Free Group Factors, 163 3.5 Graph Operators Induced by Regular Trees, 167 3.5.1 Graph Hilbert Space Generated by Regular Trees, 169 3.5.2 Representations of N-Tree Operators on Vertex Spaces, 171 3.6 Bibliography, 196

4. Fractals on Graph Groupoids 199 4.1 Introduction, 199 4.1.1 Automata and Fractal Groups, 200 4.1.2 Right Graph Von Neumann Algebras, 203 4.1.3 M-Valued Right Graph W*-Probability Spaces, 206 4.2 Labeled Graph Groupoids and Graph Automata, 209 4.3 Graph Fractaloids, 210 4.4 Labeling Operators of Graph Fractaloids, 218 4.4.1 Labeling Operators, 218 4.4.2 Free Distributional Data of Labeling Operators, 220 4.4.3 Labeling Operators of Graph Fractaloids, 221 4.4.4 Refinements of (M3), 225 4.5 Graph-Theoretical Characterization, 231 4.5.1 Graph Fractaloids Redefined, 231 4.5.2 Graph-Theoretical Characterization of Graph Fractaloids, 237

x Contents

4.6 Fractal Graphs Constructed by Fractal Graphs, 240 4.7 Fractal Pairs of Graph Fractaloids, 244 4.8 Equivalence Classes of Graph Fractaloids, 247 4.9 Completely Finite Fractalization, 250 4.10 Fractalized-Graph von Neumann Algebras, 258 4.11 Fractalized Labeling Operators, 264 4.12 Bibliography, 268

5. Entropy Theory on Graphs 271 5.1 Entropy, 271 5.2 Entropy on Finite Graphs, 273 5.3 Entropy of Finite Fractal Graphs, 279 5.3.1 Basic Computations, 279 5.3.2 Entropy of a Finite Fractal Graph, 284 5.4 Bibliography, 292

6. Jones Index Theory on Graph Groupoids 293 6.1 Introduction, 293 6.2 Quotient Graphs and Graph-Index, 293 6.2.1 The Quotient Graph G1:0 of G0 G1, 294 6.2.2 Basic Construction for G0 G1, 301 6.2.3 Special Case: Full-Vertex Subgraph Inclusions, 307 6.3 Watatanis Extended Jones Index Theory, 308 6.4 Index Theory on Graph von Neumann Algebras, 309 6.4.1 Index Theory for Canonical Conditional Expectations, 310 6.4.2 Finite-Index Type Finite-Graph von Neumann Algebras, 319 6.4.3 Infinite-Index Type Finite-Graph von Neumann Algebras, 320 6.4.4 Connection Between [G1 : G0] and IndE, 321 6.5 Basic Constructions Induced by Full-Vertex Subgraph Inclusions, 325 6.6 Ladders and Nets on Graph von Neumann Algebras, 338 6.6.1 Ladders of Graphs, 339 6.6.2 Vertex-Subgraph Inclusions, 342 6.6.3 Ladders Induced by Graph von Neumann Algebras, 343 6.6.4 Quadruples in a Ladder, 345 6.6.5 Nets of Graph von Neumann Algebras, 348 6.7 Index-Morphisms, 350 6.7.1 Graph-Index-Morphism, 351 6.7.2 Index-Morphism on Graph von Neumann Algebras, 353 6.7.3 Classification of Finite Trees, 356 6.8 Bibliography, 361 7. Network Theory on Graphs 363 7.1 Electric Resistance Network Theory, 363 7.1.1 Networks and Network Groupoids, 364 7.1.2 Ohms Law and ERNs, 370

Contents xi

7.2 Representations of ERNs, 373 7.2.1 Energy Hilbert Spaces, 373 7.2.2 Dissipation Hilbert Space, 377 7.3 ERN-Actions on Energy Hilbert Spaces, 380 7.3.1 Transfer Operators and Laplacians, 382 7.3.2 Energy Form and ERN-Actions on He, 384 7.4 Free Structures Induced by ERN-Groupoids, 391 7.4.1 Free-Moment Computations in (UG, ex), 391 7.4.2 Free-Cumulant Computations in (UG, ex), 395 7.5 Bibliography, 401

8. K-Theory on Graphs 404 8.1 Introduction, 404 8.2 K-Theory, 405 8.3 Projections in UG, 407 8.4 Projections in M(UG), 414 8.5 K0-Groups of UG, 420 8.5.1 K0-Group K0(UG), 420 8.5.2 Dimension Group K0(UG), 423 8.6 Bibliography, 426

Index 429

xii Contents

Chapter 1

Algebra on Graphs

In this chapter, we will study algebra on directed graphs. First, we constructcorresponding natural algebraic structures of directed graphs, which aregroupoids. Groupoids induced by graphs are said to be graph groupoids. Bystudying graph groupoids, we investigate algebra on graphs.

Next, we consider operations on graphs: union, gluing, product, quotient,pull-back process, complement on graphs. By these operations, we canconstruct new graphs from given graphs.

1.1 Introduction

In recent decades, groupoids have found a variety of uses. The defining axiomsare less restrictive than those for groups, and hence the domain of potentialapplications widens. Like with groups, they arise as transformations, i.e., asgroupoid actions. A unitary representation of a group yields a group of unitaryoperators on a Hilbert space, the corresponding notion of a representationof a groupoid leads to system of operators that only compose when certainmatching rules hold, groupoids are typically represented by partial isometries.For partial isometries as at u, the composition uv will be feasible only if thefinal projection of v matches the initial projection of u. These differencesimply that the algebras of Hilbert space operators generated by groupoidrepresentations have a more subtle nature than the corresponding theory forgroups, as pioneered by J. von Neumann, and Mackey and Zimmer. Similarly,the operations of products and twisted products (crossed products) exhibitfeatures unlike those typically encountered in the case of groups.

The study of groupoid actions and representations of groupoids includessuch areas as geometry, the study of commutative and noncommutativestructures, homotopy theory, holonomy, number theory, harmonic analysis,computations with infinite matrices, renormalization, quantum mechanics,and dynamical systems.

A graph is a set of objects called vertices (or points or nodes) connected bylinks called edges (or lines or curves). In a directed graph, the two directionsare counted as being distinct directed edges (or arcs). A graph is depictedin a diagrammatic form as a set of dots (for vertices), jointed by curves (foredges). Similarly, a directed graph is depicted in a diagrammatic form as a

1

2 Algebra on Graphs

set of dots jointed by arrowed curves, where the arrows indicate the directionof the directed edges.

More precisely, our directed graphs G are the pairs (V (G), E(G)), consistingof the vertex sets V (G) of G, and the edge sets E(G) of G, equipped with thedirections (or orientation) on G, i.e., every edge e has its initial vertex (orsource) v and its terminal vertex (or sink) v. For our purposes, all graphsare directed in this monograph. So, if there is no confusion, graphs meandirected graphs.

Recently, directed graphs and their graph groupoids have been studied indiscrete mathematics, harmonic analysis, operator algebra, dynamical systems,network theory, and quantum physics, etc. Directed graphs are interesting, inparticular, in operator algebra, since every edge of a given graph G assigns apartial isometry, and each vertex of G assigns a projection on a certain Hilbertspace. In other words, a graph G can have Hilbert-space representations.

In this chapter, we consider various types of operations on graphs. Thekey findings of this chapter can be summarized as follows.

By a minute change of notation, lets denote by o(G1, G2), a new graphafter an operation o. Similarly, lets denote by O(G1, G2) a new groupoidinduced by an operation O, where Gk are the graph groupoids of Gk, fork = 1, 2. Note that we do not know O(G1, G2) is again a graph groupoid.We only know O(G1, G2) is a (pure algebraic) groupoid. Our main theoremsof this chapter show that:

O(G1,G2) = the graph groupoid of o(G1, G2),

possibly under certain additional conditions, i.e., the operated grupoid ofgiven two graph groupoids is a graph grupoid of a certain operated graphs.

1.2 Preliminaries

In this section, we introduce basic concepts and fundamental results which weneed for our later works.

1.2.1 Graph Groupoids

Let G be a countable directed graph with its vertex set V (G), and its edgeset E(G). Since our graph G is directed, each edge e in E(G) has its initialvertex v and terminal vertex v, i.e., the edge e connects v to v. We denotethis relation by e = vev. Remark that the vertices v and v are not necessarilydistinct in V (G). For instance, if e is a loop-edge of G, then v = v in V (G).

Let e1 = v1ev1, and e2 = v2e2v2 be edges in E(G), with vk, vk V (G),for k = 1, 2. If the terminal vertex v1 of e1, and the initial vertex v2 of e2 areidentical in V (G), i.e., v1 = v2 in V (G), then we obtain the finite path e1 e2 onthe graph G, with its initial vertex v1 and its terminal vertex v2. Inductively,we can have finite paths on G. Note that all finite paths on G are finite wordsin E(G). The set of all finite paths on G is denoted by FP (G), and we call it

1.2 Preliminaries 3

the finite path set of G. Clearly, E(G) FP (G). If w FP (G), then it is afinite word e1 e2 ... en of edges e1, ..., en. In this case, the initial vertex of wis the initial vertex of e1, say v, and the terminal vertex of w is the terminalvertex of en, say v. Similar to the edge case, we denote w by w = vwv toemphasize the initial and the terminal vertices of w. We also denote

w = vw = wv = vwv.If w = e1 e2 ... en, then length |w| of w is defined to be n, the cardinality

of edges e1, ..., en, generating w.Now, let w1 and w2 be finite paths in FP (G), and suppose the terminal

vertex of w1 and the initial vertex of w2 are identical in V (G). Then we canconstruct a new finite path w by connecting w1 and w2, i.e., w = w1 w2.However, in general, the product (or the word) w1 w2 (resp., in E(G)) isundefined. Indeed, whenever the terminal vertex of w1, and the initial vertexof w2 are distinct, in V (G), we cannot define the finite path w1w2. If w1w2 isa well-defined finite path, then we say that w1 and w2 are admissible. Notethat, even though w1 and w2 are admissible, w2 and w1 are not admissible,in general. Similarly, if v is the initial vertex of w (or if v is the terminalvertex of w), then we say that v and w are admissible (resp., w and v areadmissible).

Define now a set F+(G) by

{} V (G) FP (G),set-theoretically, and define a binary operation by

(w1, w2) % w1 w2 ={

w1w2 if w1 and w2 are admissible

otherwise,for all w1, w2 F+(G). If w1 is a initial vertex of w2, then w1w2 is identifiedwith w2; if w1 and w2 are not admissible, then w1w2 is defined to be , whichmeans the empty word in V (G) E(G).Definition 1.2.1 The algebraic pair (F+(G), ) is called the free semigroupoidof G.We simply denote this pair by F+(G). The binary operation () is referredto as admissibility.

Note that there are some free semigroupoids that do not contain the emptyword. For example, if a graph G is the one-vertex-n-multi-loop-edge graph,then the free semigroupoid F+(On) does not contain the empty element ,because all edges of On are admissible from each other via the unique vertexof On. However, in general, whenever |V (G)| > 1, the empty element isalways contained in F+(G). Thus, without loss of generality, if there is noconfusion, we always assume that is contained in F+(G).

Let G be a given graph. The opposite directed graph G1 of G is a graphwith

V (G1) = V (G),

andE(G1) = {e1 : e E(G)},

4 Algebra on Graphs

wheree = vev e1 = ve1v,

for all e E(G), and e1 E(G1). This new directed graph G1 is calledthe shadow of G. By the very definition,

(G1)1 = G.

Construct a new graph G, by a graph with

V (G) = V (G) = V (G1),

and

E(G) = E(G) E(G1).This graph is said to be the shadowed graph of G. As a graph, the shadowed

graph G of G induces its free semigroupoid F+(G).Let F+(G) be the free semigroupoid of the shadowed graph G of G. Define

the reduction (RR) on F+(G), by(RR)

w1w = v and ww1 = v,

whenever w = vwv in F+(G), with v, v V (G).Definition 1.2.2 The quotient set F+(G) / (RR), equipped with the inheritedadmissibility () on F+(G), is called the graph groupoid of G. For convenience,we will denote the graph groupoid (F+(G) / (RR), ) simply by G.

For instance, the graph groupoid On of the one-vertex-n-multi-loop-edgegraph On, with

V (On) = {v}, and E(On) = {ej = vejv : j = 1, ..., n},for n N, is a group; moreover, it is group-isomorphic to the free group Fnwith n-generators.

1.2.2 Groupoids and Graphs

We say an algebraic quadruple (X , Y, s, r) is a (categorial) groupoid, if itsatisfies the conditions: (i) Y X, (ii) for all x1, x2 X , there exists apartially-defined binary operation (x1, x2) % x1 x2 depending on the sourcemap s and the range map r satisfying the followings;

(ii-1) if x X , then there exist y, y Y, such thats(x) = y and r(x) = y,

satisfying x = y x y,(ii-2) x1 x2 is well-determined, whenever

r(x1) = s(x2),

1.2 Preliminaries 5

and in this case,

s(x1 x2) = s(x1) and r(x1 x2) = r(x2),

for x1, x2 X ,(ii-3) (x1 x2) x3 = x1 (x2 x3) in X , if x1x2 and x2x3 are well-determined

in the sense of (ii-1), for x1, x2, x3 X ,(ii-4) if x X , then there exists a unique element x1 for x satisfying x

x1 = s(x) and x1 x = r(x). In particular, we call x1, the groupoid inverseof x, for all x X .

The subset Y of a grupoid X = (X , Y, s, r) is said to be the base of X .Every group is automatically a groupoid (X , Y, s, r) with

|Y| = 1, and hence s = r on X .Now, we put the empty element in a groupoid X , if |Y| 2. The empty

element signifies the undefinedness of the binary operation on X . By putting, we make the partially-defined operation on X be a well-defined operation.So, if there is no confusion, we assume that the empty element is containedin X .

It is easy to check that every graph groupoid G of a graph G is a groupoidwith its base V (G), i.e.,

G = (G, V (G), s, r),

where

s(w) = v and r(w) = v,for all w = vwv G, with v, v V (G).

Let Xk = (Xk, Yk, sk, rk) be groupoids, for k = 1, 2. We say that a mapf : X1 X2

is a groupoid morphism, if

(i) f is a well-defined function,

(ii) f(Y1) Y2,(iii) s2 (f(x)) = f (s1(x)) in X2, for all x X1, and(iv) r2 (f(x)) = f (r1(x)) in X2, for all x X1.If a groupoid morphism f is bijective, then we say that f is a groupoid-

isomorphism, and the groupoids X1 and X2 are said to be groupoid-isomorphic.Recall that two countable directed graphsG1 andG2 are graph-isomorphic,

via a graph-isomorphism

g : V (G1) E(G1) V (G2) E(G2),if

(i) g is bijective from V (G1) onto V (G2),

(ii) g is bijective from E(G1) onto E(G2),

6 Algebra on Graphs

(iii) g(e) = g(v1ev2) = g(v1) g(e) g(v2) in E(G2),

for all e = v1ev2 E(G1), with v1, v2 V (G1).It is not difficult to show that if two graphsG1 andG2 are graph-isomorphic,

then the corresponding graph groupoids G1 and G2 are groupoid-isomorphic.More generally, we have that:

Proposition 1.2.1 If two graphs G1 and G2 have graph-isomorphic shadowedgraphs G1 and G2, then G1 and G2 are groupoid-isomorphic.

Proof. Let Gk be the shadowed graphs of Gk, and let Gk be the graphgroupoids of Gk for k = 1, 2. Assume that they are graph-isomorphic, via agraph-isomorphism

g : V (G1) E(G1) V (G2) E(G2).Construct a grupoid morphism

: G1 G2by

(w)def=

g(w) if w V (G1) E(G1)g(e1)g(e2)...g(ek) if w = e1e2...ek, with k > 1 if w = ,in G2, for all w G1. Then is a well-defined bijection, satisfying

(V (G1)

)= g

(V (G1)

)= V (G2),

and

(w) =( v1wv2) = g(v1)(w)g(v2),

in G2, for all w = v1wv2 G1, with v1, v2 V (G1). Therefore, is agroupoid-isomorphism, equivalently, the graph grupoids G1 and G2 aregroupoid-isomorphic. !

Let X = (X , Y, s, r) be a groupoid. We say that this groupoid X actson a set Y, if there exists a groupoid action pi of X acting on Y, in the sensethat:

(i) pi(x) : Y Y is a well-determined function, for all x X , and(ii) pi(x1x2) = pi(x1) pi(x2), for all x1, x2 X ,

where () means the usual functional composition. In this case, the set Y iscalled a X -set.

Let X1 be a subset of a groupoid X2 = (X2, Y2, s, r). Suppose X1 = (X1,Y1, s, r) is again a groupoid, where Y1 = X1 Y2. Then the groupoid X1 issaid to be a subgroupoid of X2. Clearly, all subgroups are subgroupoids.

We will say that a graph G1 is a part of G2, if G1 is a full-subgraph, or asubgraph, or a vertex subgraph of G2.

Recall that a graph G1 is a full-subgraph of G2, if G1 is a graph with

1.2 Preliminaries 7

E(G1) E(G2),and

V (G1) = {v, v V (G2) : e E(G1) s.t., e = vev}.Also, we say that G1 is a subgraph of G2, if

V (G1) V (G2),and

E(G1) = {e E(G2) : v, v V (G1) s.t., e = vev}.Finally, we say G1 is a vertex subgraph of G2, if

V (G1) V (G2), and E(G1) = ,where means the empty set. So, if we say G1 is a part of G2, then G1 is afull-subgraph, or a subgraph, or a vertex subgraph of G2.

Proposition 1.2.2 Let G1 be a part of G2. Then the graph groupoid G1 ofG1 is a subgroupoid of the graph groupoid G2 of G2.

The proof is straightforward. !

1.2.3 More about Graph Groupoids

Let X be a subset of a given graph groupoid G, and assume that X isself-shadowed (or self-adjoint), in the sense that:

X = X1, in G,

where

X1 def= {w1 : w X}.Equivalently, the subset X is decomposed by

X = X0 X10 ,for some X0 " X. Remark that the choice of X0 in X is not uniquelydetermined, however, if we fix X0 in X, then X

10 is uniquely determined

up to the choice of X0.Construct now a subset GX of G, consisting of all reduced words only in

X, where the admissibility and the reduction are inherited from those of G.

Definition 1.2.3 Let X be a self-shadowed subset of G, and let GX be asubset of G, consisting of all reduced words in X, equipped with the inheritedadmissibility from G. Then we call GX = (GX , ), the subgroupoid of Ggenerated by X.

8 Algebra on Graphs

It is easy to verify GX is indeed a subgroupoid of G. However, we cannotguarantee GX is a graph groupoid.

Moreover, in general, GX is not a graph groupoid induced by a part ofG.Let Kn be the one-flow circulant graph with

V (Kn) = {v1, ..., vn}and

E(Kn) = {e12, e23, ..., en1, n, en,1},where eij means the edge connecting a vertex vi to a vertex vj . Take X = {w,w1} in the graph groupoid Kn of Kn, where

w = e12 e23 ... en1, n en,1.

Then the subgroupoid Kn:X generated byX can never be a graph groupoidinduced by a part of Kn. But, we may understand Kn:X as a graph groupoidinduced by a certain graph (which is not a part of Kn)! Lets construct agraph Kn:X by a graph with

V (Kn:X) = {v1} V (Kn),and

E(Kn:X) = {w} FPr(Kn),where Kn is the shadowed graph of Kn. Then Kn:X is graph-isomorphic tothe one-vertex-one-loop-edge graph O1,

v1!w.

Thus, subgroupoid Kn:X of the graph groupoid Kn of Kn is groupoid-isomorphic to the graph groupoid O1 of O1.

As we discussed in the previous example, we can obtain the followingproposition.

Proposition 1.2.3 Let G be a graph with its graph groupoid G, and let Xbe a self-shadowed subset of G. Let GX be the subgroupoid of G generatedby X. Then there exists a graph GX such that the graph groupoid of GX isgroupoid-isomorphic to GX .

Proof. Let X be a given self-shadowed subset of G, and let GX be thesubgroupoid of G generated by X.Construct a new graph GX with

E(GX) = X FPr(G),and

V (GX) = {v, v V (G) : w X s.t., w = vwv}.

1.2 Preliminaries 9

Then we can choose a graph GX having its shadowed graph GX , sinceX is self-shadowed. Lets denote the graph groupoid of GX by Go. SinceGo and the subgroupoid GX shares the common generator set X, they aregroupoid-isomorphic. !

Note that the choice of GX is not uniquely determined in the proof.However, the choice of GX is unique up to shadowed-graph-isomorphisms.Thanks to above proposition, we infer that every subgroupoid of a graphgroupoid is a graph groupoid, too.

Definition 1.2.4 Let X1 and X2 be self-shadowed subsets of a given graphgroupoid G, and let GXk be the subgroupoids of G generated by Xk, for k =1, 2. Define the free product GX1 GX2 in G, as the subgroupoid GX1X2 .

Note that if there exists at least one pair (w1, w2) GX1 GX2 , suchthat w1 and w2 are admissible (or, respectively, w2 and w1 are admissible),then

GX1 GX2 # GX1 GX2 .If X1 and X2 are totally disconnected, in the sense that: for all pair

(w1, w2) of X1 X2, the elements w1 and w12 , and w11 and w12 arenot admissible in G, then

GX1 GX2 = GX1 GX2 = GX1 unionsq GX2 .Proposition 1.2.4 Let G be the graph groupoid of a graph G. Then

G Groupoid= eE(G)

Ge,

where Ge are the subgroupoids of G, generated by Xe = {e, e1}, for all e E(G). Moreover, if e is a loop-edge, then Ge is groupoid-isomorphic to theinfinite cyclic abelian group Z, and if e is a non-loop-edge, then Ge is a finitegroupoid

{, v, v, e, e1},whenever e = v e v E(G), with v 2= v V (G).Proof. By definition, the graph groupoid G of a graph G is generated bythe self-shadowed subset

E(G) = E(G) E(G1).Therefore,

G = GE(G) = G eE(G)

{e,e1} = eE(G)

G{e,e1} = eE(G)

Ge.!

The above proposition provides the algebraic characterization of graphgroupoids. Note that the free product in graph groupoids is totallydependent upon the admissibility on the graph groupoids. Therefore, wesometimes use the notation G, instead of using , to emphasize that ourfree product is acting in the graph groupoid G of G. With help of this newnotation, we can re-write the above result as follows:

G Groupoid= GeE(G)

Ge.

10 Algebra on Graphs

1.3 Groupoids under Operations

In this section, we consider the operations on the collection of groupoidsin the sense of Section 2.2. The sum, the product, the quotient, and thecomplemented groupoids are introduced and studied.

1.3.1 Sum of Groupoids

Let Xk = (Xk, Yk, sk, rk) be groupoids, for k = 1, 2. Define a new groupoidX1 + X2 by the groupoid generated by X1 and X2, i.e.,

X1 + X2 def= the groupoid generated by X1 X2.Definition 1.3.1 The groupoid X1 + X2, generated by the groupoids X1 andX2, is called the groupoid sum of X1 and X2. If X1 X2 = , then we denoteX1 + X2 by X1 X2. This sum X1 X2 is called the direct (groupoid) sumof X1 and X2.

Clearly, by the operation on the groupoid sums, if we have direct sumX1 X2, then x1 x2 = , whenever x1 X1 (resp., x1 X2) and x2 X2(resp., x2 X1). Thus, we obtain that

X1 X2 = X1 unionsq X2,set-theoretically, where unionsq means the disjoint union.

Let k = < Xk, Rk > be finitely presented groups with the generator setsXk and the relations Rk, for k = 1, 2. The groupoid sum 1 + 2 is again afinitely presented group,

1 + 2 = < X1 X2, R1 R2 > .Clearly, if X1 X1 = = R1 R2, then we obtain the group direct sum

1 2 of 1 and 2.

1.3.2 Product Groupoids

Let Xk = (Xk, Yk, sk, rk) be groupoids, for k = 1, 2. Define the productgroupoid X = X1 X2 by

X1 X2 = (X1 X2, Y1 Y2, s, r),where

X Y def= {(x, y) : x X, y Y }means the Cartesian product of arbitrary sets X and Y , and where s and rare the source and the range maps from X1 X2 onto Y1 Y2, defined by

s ((x1, x2))def= (s1(x1), s2(x2)) ,

1.3 Groupoids under Operations 11

and

r ((x1, x2))def= (r1(x1), r2(x2)) ,

for all (x1, x2) X1 X2. Clearly, the maps s and r are onto, since sk and rkare onto, for k = 1, 2. Of course, the binary operation on X1 X2 is definedby

(x1, x2) (x1, x2)def= (x1 x1, x2 x2),

with identity:

(, x2x2) = = (x1x1, ),for all (x1, x2), (x1, x2) X1 X2, where xk xk means the product of xkand xk in Xk, for k = 1, 2.

All product groups are automatically product groupoids. For example, letk = < Xk, Rk > be finitely presented groups, for k= 1, 2. Then the productgroup 1 2 is defined again by the finitely presented group,

1 2 = < X1 X2, R1 R2 > .Let G1 and G2 be graph groupoids. Since both G1 and G2 are groupoids,

we can construct the product groupoid G1 G2. Is this new groupoid G1 G2 a graph groupoid?

1.3.3 Quotient Groupoids

In this section, we define the quotient groupoids. Let Xk = (Xk, Yk, sk, rk)be groupoids, for k = 1, 2. Assume that X2 is a subgroupoid of X1, satisfyingthe following conditions:

(i) X2 X1,(ii) Y2 = Y1, and(iii) s2 = s1 |X2 , and r2 = r1 |X2 .By (iii), without loss of generality, we denote

Xk = (Xk, Yk, s, r), for k = 1, 2.Note now the condition (ii). We assumed the bases Yk of Xk are identical

(or bijective, or equipotent). In fact, we can replace the condition (ii) by themore general condition:

Y2 Y1,and we may put certain additional conditions to define the quotient groupoidX1/X2. However, for convenience, we restrict our interests to the case wherethe groupoids X1 and X2 have the same bases Y1 and Y2, respectively. So, wemay denote


Xk = (Xk, Y, s, r), for k = 1, 2.Suppose now that such a subgroupoid X2 of X1 satisfies the following condition,

x X2 x1 X2, in X1, for all x X1.Then we have an equivalence relation R on X1, defined by

x1 R x2 def x1x12 X2,for x1, x2 X1. Indeed, the relation R is an equivalence relation, because:(i) x R x, for all x X1:

For any x X1, x x1 Y X2.(ii) x1 R x2 = x2 R x1, for x1, x2 X1:

Assume that x1 R x2. Equivalently x1x12 X2. Then the groupoidinverse (

x1x12

)1= x2 x

11

of x1x12 is also contained in X2, and hence x2 R x1.

(iii) x1 R x2 and x2 R x3 = x1 R x3.Let x1 R x2, and x2 R x3, i.e., assume that x1x12 , x2x13 X2 in X1.Then the product (x1x

12 )(x2x

13 ) is identical to x1x

13 , or .

So, x1 R x3.So, the relation R is an equivalence relation. Thus, we can construct the

equivalence classes in X1 by

[x]def= {a X1 : x R a} X1.

Then, the groupoid X1 is decomposed byX1 =

xX1[x],

set-theoretically. Now, we can define a quotient set

X1/R def= {[x] : x X1}.On this quotient set, define the operation () by

[x1] [x2]def= [x1x2], for x1, x2 X1.

Then the algebraic structure (X1 / R, ) is again a groupoid.

Definition 1.3.2 The above groupoid (X1/R, ) is called the quotient groupoidof X1 by X2. And we denote this groupoid simply by X1/X2.

1.3 Groupoids under Operations 13

How about the general case where Y2 Y1, where Yk are the bases of Xk, fork = 1, 2? To consider this general case, we need one more concept, called thepull-back process.

Let X = (X , Y, s, r) be a groupoid and let Y be an arbitrary set, satisfyingthat |Y | |Y| . Determine a surjection

g : Y Y.Clearly, the choice of g is not uniquely determined. There can be many

different surjections. Assume that we fix one of them, g. Then we can definea groupoid Xg(Y ) = (Xg(Y ), Y, sY , rY ), containing Y as its base, by thebijection

g : X \ Y Xg(Y ) \ Y,satisfying

sY (g(x)) = g (s(x)) , and rY (g(x)) = g (r(x)) ,

for all x X \ Y. Define now a mapg0 : X Xg(Y )

by

g0(x)def=

{g(x) if x Yg(x) if x X \ Y .

Then, the map g0 is a surjection, since g is surjective and g is bijective.Set-theoretically, we obtain that:

Xg(Y ) = g0(X ).If there is no confusion, then we denote sY and rY simply by s and r,

respectively, i.e., for the surjection g : Y Y, we can determine a newgroupoid Xg(Y ).Proposition 1.3.1 Let Xg(Y ) = (Xg(Y ), Y, s, r) be a groupoid induced by agroupoid X = (X , Y, s, r), where g : Y Y is a surjection, as above. ThenXg(Y ) is indeed a groupoid in the sense of Section 2.2.Proof. Recall that Xg(Y ) = g0(X ), where g0 is defined as above. Checkthat the set g0(X ) satisfies the following conditions:

(1) For x g0(X ), there exist the source s(x) and the range r(x). Indeed,if x Y, then s(x) = x = r(x); and if x g0(X ) \ Y, then

s(x) = s(g10 (x)

)= s

((g)1(x)

),

and

r(x) = r(g10 (x)

)= r

((g)1(x)

),

since g is bijective.


(2) For x1 and x2 in g0(X ), we can define the product x1x2 in g0(X ), bythe rules;

x1x2 = x1 r(x1) s(x2) x2 =

{x1x2 if r(x1) = s(x2) otherwise,

by (1). It suffices to show that Xg(Y ) satisfies (1) and (2), to prove Xg(Y ) =g0(X ) is a groupoid. !

The above proposition shows that, indeed Xg(Y ) = (g0(X ), Y, s, r) is agroupoid.

Definition 1.3.3 The new groupoid Xg(Y ) = (g0(X ), Y, s, r) induced by agroupoid X = (X , Y, s, r) and a surjection g : Y Y, for a fixed set Y (|Y | |Y|) is called the g-pull-back groupoid of X . The surjection g is called thepull-back map. Also, the construction of Xg(Y ) from X is called the pull-back(process) of X by g. Remark here that the pull-back groupoids Xg(Y ) of a fixedgroupoid X is completely determined by the pull-back maps g, even though wehave a fixed set Y.

Now, let X2 X1 be groupoids given as above, without the condition (ii).Assume that the base Y2 of X2 satisfies only

(ii) Y2 Y1.i.e., we replace the condition (ii) by condition (ii). Then we can define thepull-back map

g : Y1 Y2and we can construct the g-pull-back groupoid (X1)g(Y2), where thecorresponding bijection g satisfies

g(x) = x in (X1)g(Y2), for all x X1,equivalently,

(X1)g(Y2) = g0 (X1) ,where g0 is defined like above. Then we can derive that:

Proposition 1.3.2 Let X2 X1, and (X1)g(Y2) be given as in the aboveparagraph. Then the subgroupoid X2 of X1 is again a subgroupoid of the g-pull-back back groupoid (X1)g(Y2) of X1.The proof is trivial by the very definition. "

Consider X2 as a subgroupoid of the g-pull-back groupoid (X1)g(Y2) (bythe above proposition). Then we can give an equivalence relation Rg by

x1 Rg x2 def (g0(x1)) (g0(x2))1 X2,for x1, x2 X1. Then, we can re-define the (general) quotient groupoids asfollows:

1.4 Operations on Graphs 15

Definition 1.3.4 Let Xk = (Xk, Yk, s, r) be groupoids and assume thatX2 X1, satisfy the conditions (i), (iii), and (ii). Then the quotient groupoidX1/X2 is defined by the groupoid (X1/Rg, Y2, s, r), where X1/Rg is thequotient set consisting of all equivalence classes generated by the equivalencerelation Rg, on X1.Assumption. However, in our later context, when we mention quotientgroupoids X1/X2, we will assume X2 satisfy the conditions (i), (ii), and (iii)introduced at the beginning of this section, without considering the pull-backprocess. "

1.3.4 Complemented Groupoids

Now, let Xk = (Xk, Yk, sk, rk) be groupoids, for k = 1, 2. Define a newgroupoid X1 X2 of X2 in X1, by

X1 X2 def= ((X1 \ X2) D1:2, D1:2, s, r) ,where

D1:2 def= {s(x), r(x) : x X1 \ X2} .Then it is a new groupoid.

Definition 1.3.5 The new groupoid

X1 X2 = ((X1 \ X2) D1:2, D1:2, s, r) ,is called the complemented groupoid of X2 in X1. If X2 X1, then we denotethe complemented groupoid X1 X2 by X c2 , like in set theory.

Notice that, in general,

Y1 \ Y2 D1:2,where Yk are the bases of Xk, for k = 1, 2. The equality holds, only when X2is totally disconnected, in the sense that: for all pair,

(x1, x2) (X1 \ X2) X2,the products x1x2 = in X1.

1.4 Operations on Graphs

In this section, we consider the various operations on graphs. We will showthat a groupoid X induced by an operation on two graph groupoids G1 andG2 is groupoid-isomorphic to a graph groupoid G of a certain graph G,induced by a suitable operation on the graphs G1 and G2, generating G1 andG2, respectively. It means that the study of a new operated groupoid X isto investigate another graph groupoid G of a certain graph G, induced by G1and G2.

Our observation provides a way to study operated groupoids of graphgroupoids, in terms of graph groupoids.


1.4.1 Unioned Graphs

Let Gk be fixed graphs with their graph groupoids Gk, for k = 1, 2. Weconsider the union G = G1 G2 of G1 and G2, and corresponding groupoidal,and dynamical structures. In conclusion, we show that the graph groupoid Gof G is the sum G1 + G2 (in the sense of Section 4.1) of the graph groupoidsG1 and G2. Thus we conclude that the graph von Neumann algebra MGis -isomorphic to the D1:2-valued free product algebra MG1 rD1:2 MG2 , ofgraph von Neumann algebras MG1 and MG2 , where D1:2 is the commonW -subalgebra DG1 DG2 of the diagonal subalgebras DGk of DGk , fork = 1, 2.

1.4.1.1 The Union on Graphs

Let Gk be graphs with graph groupoids Gk, for k = 1, 2. We define a newgraph G1 G2 constructed by the given graphs G1 and G2.Definition 1.4.1 Let Gk be given, for k = 1, 2. Define a new graph G by agraph with

V (G) = V (G1) V (G2),and

E(G) = E(G1) E(G2).Then this new graph G is called the unioned graph of G1 and G2, and we

denote G by G1 G2, whenever we want to emphasize the originally givengraphs G1 and G2.

Note that our unioned graphs are different from the disjoint unionedgraphs. In fact, every disjoint unioned graph is a unioned graph, but not allunioned graphs are disjoint unioned graphs. To distinguish the disjoint unionand our (general) union on graphs, we denote G1 unionsq G2, for the disjoint unionof G1 and G2, i.e., G1 unionsq G2 is the graph with

V (G1 unionsqG2) = V (G1) unionsq V (G2),and

E(G1 unionsq G2) = E(G1) unionsq E(G2),where unionsq on the right-hand sides means the disjoint union (set-theoretically).

The best example to illustrate that, unioned graphs are not disjoint unionedgraphs is the shadowed graphs. Let G be a graph and let G1 be the shadowof G. Then they share the vertex sets, i.e., V (G) = V (G1). So, the shadowedgraph G of G (or of G1) is the unioned graph G G1, with

V (G) = V (G) V (G1) = V (G) = V (G1),


and

E(G) = E(G) E(G1) = E(G) unionsq E(G1).Note that if G is connected, then the shadowed graph G = G G1 is

connected, too. So, it also illustrate that our unioned graphs are not disjointunioned graphs, in general.

Also, let K be a sufficiently big graph containing subparts as, G1 and G2.And assume that either

V (G1) V (G2) 2= or E(G1) E(G2) 2= ,where means the empty set. Then we can construct a new subpartG = G1 G2 of K. And, in such a case,

G 2= G1 unionsq G2,too.

Now, recall that if Xk are groupoids, then we can derive the sum X1 + X2of X1 and X2, as a new groupoid equipped with the binary operation. Thefollowing theorem shows that the sum G1 + G2 of graph groupoids G1 andG2 is again a graph groupoid, moreover G1 + G2 is groupoid-isomorphic tothe graph groupoid G of the unioned graph G = G1 G2.

Theorem 1.4.1 Let Gk be graphs with their graph groupoids Gk, for k = 1,2. If G = G1 G2 is the unioned graph of G1 and G2, then the graph groupoidG of G is groupoid-isomorphic to the sum G1 + G2 of the graph groupoids G1and G2.

Proof. Let G = G1 G2 be the unioned graph of G1 and G2. Then, asa new directed graph, G has its own graph groupoid G. This groupoid G isgroupoid-isomorphic to the reduced free product groupoid G

eE(G)Ge, where

Ge are the subgroupoids of G, consisting of all reduced words only in {e, e1},for e E(G). Thus, we obtain:

G Groupoid= GeE(G)

GeGroupoid

=

(G1

eE(G1)(G1)e

)G(

G2xE(G2)

(G2)x)

Groupoid= G1 G G2.

Let V1:2 be the intersection of V (G1) and V (G2), i.e.,

V1:2def= V (G1) V (G2).

Then the above formula can be re-written as follows:

G Groupoid= G1 GV1:2 G2,


where GV1:2 means that every pair (w1, w2) G1 G2 of nonempty elementscan generate a new nonempty element w1w2 in G, if and only if the terminalvertex of w1, denoted by r(w1), and the initial vertex of w2, denoted by s(w2)are identical in V1:2. Indeed, it is not difficult to show that r(w1) = s(w2) inV (G), if and only if they are identical inside V1:2, whenever wk Gk \ {},for k = 1, 2.

This V1:2-amalgamated reduced free product G1 GV1:2 G2 shows theadmissibility on G. Thus, we can define a map from the graph groupoidG of the unioned graph G = G1 G2 to the sum G1 + G2 of G1 and G2,

g : G Groupoid= G1 GV1:2 G2 G1 + G2by

g(w)def=

w if w Gk, for k = 1, 2w1w2 if w = w1w2 2= , for (w1, w2) G1 G2 otherwise,for all w G. Then it is a well-defined function, moreover it is bijective,since it preserves the generators G1 G2 of G to the generators G1 G2 ofG1 + G2. And the admissibility on G, is preserved in the equivalent binaryoperation on G1 + G2, by the bijection g. Therefore, this map g is in fact agroupoid-isomorphism. Indeed, g satisfies

g (w1w2) = g(w1) g(w2), in G1 + G2,

for all w1, w2 G. !The above theorem shows that the groupoid G1 +G2, the sum of the

graph groupoidsG1 ofG1 and G2 ofG2, is groupoid-isomorphic to the graphgroupoid G of the graph union: G = G1 G2. Thus, it provides a way tostudy the sum G1 + G2 by investigating the graph groupoid G of G = G1 G2. In short, the new groupoid, the sum G1 + G2 of graph groupoids G1 andG2, is again a graph groupoid.

By the previous theorem, we obtain the following corollary.

Corollary 1.4.1 Let Gk be the graph groupoids of graphs Gk, for k = 1, 2.The direct sum G1 G2 of G1 and G2 is groupoid-isomorphic to the graphgroupoid G of the disjoint graph union G = G1 unionsq G2. "

Inductively, we obtain the following corollary.

Corollary 1.4.2 Let Gk be the graphs with their graph groupoids Gk, fork = 1, ..., N, for N N \ {1}, and let X be the sum

X def= Nj=1 Gj = G1 + G2 + ... + GN .Then:

(1) X Groupoid= G1 GVN G2 GVN GVN ... GVN GN =N

GVNj=1

Gj , where GVN isthe VN -free product in X , and


VNdef=

(k,l){1,...,N}2Vk,l,

where Vk,l = V (Gk) V (Gl).(2) if G =

Nj=1

Gj is the iterated graph union of G1, ..., GN , then the graph

groupoid G of G is groupoid-isomorphic to X ."

Lets consider the following two fundamental examples.

Example 1.4.1 Let Onk be the one-vertex-nk-loop-edge graphs, for nk N,for k = 1, 2. Then the graph groupoids Onk are groups, which are group-isomorphic to the free groups Fnk with nk-generators, for k = 1, 2.

Case 1 Assume that

V (On1) V (On2) = {v} = V (On1) = V (On2),and

E(On1) E(On2) = {ei1 , ..., eik},for k n1 + n2 in N. Then the graph union On1 On2 is graph-isomorphicto the one-vertex-(n1 + n2 k)-loop-edge graph On1+n2k. Thus, the graphgroupoid On1+n2k is a group again, which is group-isomorphic to the freegroup Fn1+n2k, i.e.,

O Group= On1 GC On2 Group= On1+n2k Group= Fn1+n2k,where O = On1 + On2 is the sum of On1 and On2 , and where G = On1 On2 .

Case 2 Assume now that

E(On1) E(On2) = .Then the sum O = On1 + On2 satisfies

O Groupoid= On1 GC On2 Groupoid= On1+n2 Group= Fn1+n2 ,since k = 0, in this case.

Case 3 Finally, suppose

V (On1) V (On2) = = E(On1) E(On2).Then the sum O of On1 and On2 satisfies

O Groupoid= On1 On2 Groupoid= Fn1 Fn2 ,as a graph groupoid. Notice here that in the far-right-hand side ofthe above formula means the direct sum on groupoids (not on groups),i.e., Fn1 Fn2 is not a group, because it has multi-units {i1, i2}, where ikare the group identities of Fnk , for k = 1, 2.


Example 1.4.2 Let L be a graph,

v1 e1v2e2 v3 ,

and T be a graph,

v2 e2 v3e3 v4

.

Then the corresponding graph groupoids L and K are

L = {, v1, v2, v3 e11 , e12 , (e1e2)1},and

K = {, v2, v3, v4, e12 , e13 , (e12 e3)1},set-theoretically. Then the sum X = L + K of L and K, as a groupoid,is groupoid-isomorphic to the graph groupoid G of the unioned graphG = L K, which is graph-isomorphic to

v1 e1 v2e2 v3e3 v4 .

Since X is groupoid-isomorphic to G,

X ={ , v1, v2, v3, v4, e11 , e12 , e13 ,

(e1e2)1, (e12 e3)1, (e1e3)1

}.

Now, let Gk be given as above, for k = 1, 2, and let G be the unionedgraph G2 G1. Then, by the very definition, the unioned graph G is graph-isomorphic to the unioned graph G = G1 G2. This guarantees the followingproposition.

Proposition 1.4.1 Let X = G1 + G2 be the sum of graph groupoids G1 andG2. Then X is groupoid-isomorphic to G2 + G1.

Proof. Let X =G1 +G2 be the sum of the graph groupoidsGk of graphsGk.We know that the groupoid X is groupoid-isomorphic to the graph groupoidG of the unioned graph G = G1 G2. And, by definition, it is clear thatthe unioned graph G is graph-isomorphic to the unioned graph G = G2 G1. Since the graph groupoid G of G is groupoid-isomorphic to the sumG2 + G1, we can conclude that

X = G1 + G2 Groupoid= G Groupoid= G Groupoid= G2 + G1!


1.4.1.2 Glued Graphs

In this section, we will consider a special kind of graphs unions, called theglued graphs.

Let G1 and G2 be graphs with their graph groupoids G1 and G2,respectively. Fix vertices vk V (Gk), for k = 1, 2. Then identify v1 andv2 to an ideal vertex v#. The identification process of v1 and v2 is called thegluing of v1 and v2, and the resulting vertex v# is called the glued vertex ofv1 and v2. From this gluing, we can construct a new graph G with

V (G) = {v#} (V (G1) \ {v1}) (V (G2) \ {v2}) ,and

E(G) = E(G1) E(G2),with the identification rule (I): if either ek = ekvk or ek = vkek in E(Gk), fork = 1, 2, then this edge ek is identified with an edge of G, also denoted byek, satisfying ek = ek v#, respectively, ek = v# ek in E(G).

Definition 1.4.2 Let G1 and G2 be given as above and let v# be the gluedvertex of fixed vertices vk V (Gk), for k = 1, 2. Then the above graph G iscalled the glued graph of G1 and G2, by gluing v1 and v2. To emphasize weglue v1 and v2, sometimes, we denote the glued graph G by

G1 v1#v2 G2.

After the gluing v1 and v2, we can construct two graphs G#1 and G

#2 , such

that:

V (G#1 ) = {v#} (V (G1) \ {v1}) , and E(G#1 ) = E(G1),and

V (G#2 ) = {v#} (V (G2) \ {v2}) , and E(G#2 ) = E(G2),where = means the set equality =, satisfying the identification rule (II): ifek = ek vk, or ek = vkek in E(Gk), for k = 1, 2, then this edge ek is identifiedwith an edge of G#k , also denoted by ek, satisfying ek = ek v#, respectively,

ek = v# ek in E(G#k ), for k = 1, 2.

Under the above setting, we can obtain the following proposition.

Proposition 1.4.2 Let G = G1 v1#v2 G2 be the glued graph of G1 and G2by gluing v1 and v2. Then:

(1) The graph groupoid G of G is groupoid-isomorphic to the graph groupoidG(G#) of the unioned graph G# = G#1 G#2 .

(2) The graph groupoid G of G is groupoid-isomorphic to the sum G#1 +G#2 of graph groupoids G

#k of G

#k , for k = 1, 2.

(3) The graph groupoid G of G is groupoid-isomorphic to G#1 G{v#} G#2 .


Proof. It suffices to prove statement (1). Then automatically the statements(2) and (3) are proved. And the statement (1) is trivial, by the very definitionof G#k , for k = 1, 2. Indeed, the graph G is graph-isomorphic to the unioned

graph G# = G#1 G#2 . Therefore, the graph groupoid G of G is groupoid-

isomorphic to the graph groupoid G(G#) of the unioned graph G#. !

1.4.2 Product Graphs

The main purpose of this section is to study the product groupoids of graphgroupoids. We will show that the product groupoid induced by two graphgroupoids is also a graph groupoid. Inductively, we can verify that the productgroupoid induced by finitely many graph groupoids also graph groupoids.

This shows that the study of von Neumann algebras, generated by productgroupoids of graph groupoids, is to investigate other graph von Neumannalgebras. Since we already know the structure theorems for graph vonNeumann algebras, it is not difficult to analyze the structures of product-graph-groupoid von Neumann algebras.

1.4.2.1 Vertex-Product Graphs

Throughout this section, let G1 and G2 be connected directed graphs. Forthe given two graphs G1, and G2, construct a new graph G by a graph withits vertex set V (G),

V (G)def= V (G1) V (G2),

and its edge set E(G), determined by the following edge property (E):

(E) If (v1, v1) and (v2, v2) be vertices in V (G), then there exists anedge (e, e) E(G), only if there exist edges e = v1ev2, in E(G1),and e = v1 e v2, in E(G2).

i.e., the edge set E(G) of G is determined by

E(G)def=

{(e, e)

e E(G1), e E(G2),(e, e) satisfies (E)}.

Note that the graph G is dependent upon the admissibility on both G1and G2.

Definition 1.4.3 Let G1 and G2 be connected graphs, and let G be a graphdefined as in the previous paragraph. Then this graph G is called the vertex-product graph of G1 and G2. By G1 V G2, we denote this graph G, wheneverwe want to emphasize that G is induced by G1 and G2.

By construction, we can conclude that,

E(G) E(G1) E(G2),


in general. If the graphs G1 and G2 are complete (directed) graphs, thenthe equality holds, but, in general, the equality does not hold. Recall that adirected graph G is complete, if (i) G has no loop-edges, and (ii) for any pair(v1, v2) of distinct vertices, there always exists a unique edge e of G, suchthat e = v1 e v2. For instance, the following graph G

G = #

is a complete graph with 3-vertices.Let Gk be the graph,

vk1

ekvk2,

for k = 1, 2. Then the product graph G = G1 V G2 is a graph withV (G) = {(v11, v21), (v11, v22), (v12, v21), (v12, v22)},

and

E(G) = {(e1, e2)},i.e., G is the disconnected graph

G =

(v11,v21) (v11,v22)

(v12,v21) (v12, v22).

The above example shows that, even though the given graphs G1 andG2 are connected, the vertex-product graph G1 V G2 is disconnected, ingeneral.

Consider now the graph groupoid G of the vertex-product graphG = G1 V G2. Notice here that the empty element of G are identifiedwith

(1, w2) = = (w1, 2),for all wk Gk, where k means the empty element of Gk, for k = 1, 2.

1.4.2.2 Edge-Product Graphs

Recall now that if G is the graph groupoid of a (connected) graph G, then Gis the collection of all reduced words in the edge set E(G) of the shadowedgraph G of G, under the reduction (RR) on G, and the admissibility on G (oron G). We can infer the edge set E(G), as the generator set of G. Thus, if weconstruct the suitable generator set from E(G1) and E(G2), we can investigatethe product groupoid G1 G2, in terms of a certain graph groupoid.

Let Gk be connected directed graphs with their graph groupoids Gk, fork = 1, 2. Then we can construct a new directed graph G with its edge set,


E(G)def= E(G1) E(G2)= {(e1, e2) : e1 E(G1), e2 E(G2)}.

The vertex set V (G) is the collection of all pairs (v1, v2) V (G1) V (G2),satisfying the vertex property (V):

(V) (v1, v2) V (G) V (G1) V (G2), if and only if there exists an edge(e1, e2) E(G), such that

e1 = v1e1, or e1 = e1 v1,

and

e2 = v2 e2, respectively e2 = e2 v2.

In other words, (v1, v2) V (G), if and only if there exists an edge (e1, e2) E(G), such that:

(e1, e2) = (v1, v2) (e1, e2) = (v1e1, v2e2),

or

(e1, e2) = (e1, e2) (v1, v2) = (e1v1, e2v2).

i.e., the vertex set V (G) of G is defined by

V (G)def=

{(v1, v2)

vk V (Gk), for k = 1, 2,(v1, v2) satisfies (V)}.

By the very definition,

V (G) V (G1) V (G2).Now, take (v1, v2) V (G). Then, by the connectedness of the graphs G1

and G2, there exists at least one pair (e1, e2) E(G), such that(v1, v2) = (e1e

11 , e2e

12 ), or (e1e

11 , e

12 e2), or

e11 e1, e2e12 ), or (e

11 e1, e

12 e2),

under the reduction (RR) on the graph groupoid G of G. This shows that (v1,v2) satisfies (V). Thus,

V (G) V (G1) V (G2).Remark that, in general, the equality does not hold (See the examples

below).

Definition 1.4.4 The graph G, defined above from the connected graphs G1and G2, is called the edge-product graph of G1 and G2. We denote G byG1 G2, whenever we emphasize that G is induced by G1 and G2.For example, let G1 be a graph with

V (G1) = {v1, v2, v3}, and E(G1) = {e12, e13},


where eij means the edge connecting a vertex vi to a vertex vj , i.e.,

G1 =

v2e12

v1 e13 v3,

and G2, a graph with

V (G2) = {x1, x2}, and E(G2) = {f12},where fij means the edge connecting a vertex xi to a vertex xj , i.e.,

G2 = x1 f12 x2 .Then the edge-product graph G = G1 G2 is a graph with

E(G) = {(e12, f12), (e13, f12)},and

V (G) = {(v1, x1), (v2, x2), (v3, x2)},i.e.,

G =(v2,x2)

(v1,x1) (v3,x2)

.

Note that, in this case, G is graph-isomorphic to G1, and it satisfies

V (G) " V (G1) V (G2).Now, let

Gk = vk1 ek vk2 ,for k = 1, 2. Then the edge-product graph G = G1 G2 is a graph with

E(G) = {(e1, e2)},and

V (G) = {(v11, v21), (v12, v22)},which is graph-isomorphic to G1 (and G2).Let K3 be the one-flow circulant graph with 3-vertices,

K3 =

v3e3 e2

v1 e1 v2,

and L3, the finite linear graph with 3-vertices,


L3 = x1 f1x2f2 x3 .

Then the edge-product graph G = K3 L3 is a graph withE(G) = {(e1, f1), (e1, f2), (e2, f1), (e2, f2), (e3, f1), (e3, f2)},

and

V (G) =

(v1, x1), (v2, x2), (v1, x2), (v2, x3),(v2, x1), (v3, x2), (v3, x3),(v3, x1), (v1, x3)

.Notice here that, in this case,

V (G) = V (G1) V (G2),and G is graph-isomorphic to neither G1 nor G2.

From the above three examples, we realize that by constructing the edge-product graph G = G1 G2, we can create a completely different graph fromG1 and G2. However, the graph G may contain the combinatorial informationabout the admissibility both on G1 and G2. Clearly, as a new graph, G hasits graph groupoid G.

The shadow G1 of the edge-product graph G is defined as usual. Let(e1, e2) E(G). Then the shadow (e1, e2)1 is naturally determined by

(e1, e2)1 = (e11 , e12 )

in E(G1). This relation proves the following lemma.

Lemma 1.4.1 Let G = G1 G2 be the edge-product graph of connectedgraphs G1 and G2. Then the shadow G1 of G satisfies

G1 graph= G11 G12 ,where G1k are the shadows of Gk, for k = 1, 2, and G

11 G12 is the

edge-product graph of G11 and G12 , i.e.,

(G1 G2)1 Graph= G11 G12 ."

Notice now that, even though (G1 G2)1 = G11 G12 , the shadowedgraph (G1 G2) is not graph-isomorphic to the edge-product graph G1 G2of the shadowed graphs G1 and G2!

Remark 1.4.1 Let G1 and G2 be connected graphs with their graph groupoidsG1, and G2, respectively. Then

(G1 G2) 2= G1 G2.


In general,

E(G1 G2

)" E

(G1 G2

).

Indeed, we can take an edge (e11 , e2) E(G1 G2), which is undefinedin E

(G1 G2

), where

e11 E(G11 ) E(G1) and e2 E(G2) E(G2).

Note that all edges in E(G1 G2

)have their form, either

(e1, e2) or (e11 , e

12 ),

where ek E(Gk) E(Gk), for k = 1, 2. This observation shows thatG1 G2

Full-Subgraph G1 G2.This also shows that the graph groupoid G of the edge-product graph

G1 G2 is a subgroupoid of the product groupoid G1 G2 of the graphgroupoids G1 and G2, i.e.,

GSubgroupoid G1 G2,

in general.

By the previous remark, we obtain the following proposition.

Proposition 1.4.3 Let Gk be connected graphs with their graph groupoidsGk, for k = 1, 2, and let G = G1 G2 be the edge-product graph of G1 andG2, having its graph groupoid G. Then G is a proper subgroupoid of theproduct groupoid G1 G2, whenever |E(Gk)| > 0, for k = 1, 2.Proof. As we have seen in the previous remark, the graph groupoid G ofthe edge-product graph G is a subgroupoid of the product groupoid G = G1 G2, i.e.,

GSubgroupoid G.

So, it suffices to show that G is properly contained in G, whenever E(Gk)contains at least one edge, for all k = 1, 2. Let ek E(Gk), for k = 1, 2. Thenthe groupoid G contains an element (e11 , e2). But,

(e11 , e2) / G.equivalently,

G " G.!

The above proposition shows that the graph groupoid of

G1 G2Subgroupoid

" G1 G2,in general, whenever |E(Gk)| > 0, for all k = 1, 2.


Assumption In the rest of this chapter, if we say a graph G is a productgraph of G1 and G2, then it means the graph G is the edge-product graphG1 G2.

Let G1 and G2 be connected graphs, and let G = G1 G2 be the productgraph of G1 and G2. It is natural to construct the product graph G byG2 G1. Consider now the relation between the product graphs G and G.Proposition 1.4.4 Let G = G1 G2 and G = G2 G1 be product graphs ofconnected graphs, generated by G1 and G2. Then they are graph-isomorphic.

Proof. Let G and G be given as above. It is easy to check that the edgeset E(G) and E(G) are bijective (or, equipotent) by a map

gE : E(G) E(G)defined by

(e1, e2) E(G) % (e2, e1) E(G).Therefore, there exists a natural bijection

gV : V (G) V (G)defined by

(v1, v2) V (G) % (v2, v1) V (G),because both V (G) and V (G) satisfy the vertex property (V), i.e., the vertexsets V (G) and V (G) are bijective, too, via gV .

Define now a map g : G G by gV gE , such that

g ((w1, w2))def=

{gV ((w1, w2)) if (w1, w2) V (G)gE ((w1, w2)) if (w1, w2) E(G),

in G, for all (w1, w2) V (G) E(G). Then this map is a bijective map fromV (G) E(G) onto V (G) E(G), since gV and gE are bijective. Also, itsatisfies that: for any (e1, e2) E(G),

g ((e1, e2)) = gE ((v1, v2)(e1, e2)(v1, v

2))

where ek = vk ek vk, for k = 1, 2= (e2, e1)= (v2, v1) (e2, e1) (v2, v1)= gV ((v1, v2)) gE ((e1, e2)) gV ((v1, v2))= g ((v1, v2)) g ((e1, e2)) g ((v1, v2)) .

Therefore, this bijective map g is a graph-isomorphism. !

The above theorem shows that the product graphs G = G1 G2 andG = G2 G1 are graph-isomorphic. Therefore, the graph groupoids G of G,and G of G are groupoid-isomorphic.


Proposition 1.4.5 The graph groupoid of the product graph G1 G2 ofconnected graphs G1 and G2, and the graph groupoid of the product graphG2 G1 are groupoid-isomorphic.Proof. Let

g : G = G1 G2 G = G2 G1be the graph-isomorphism introduced in the proof of the above proposition.Then we can obtain the graph-isomorphism

g : G G,where G and G are the shadowed graphs of G and G, respectively, i.e.,

g ((w1, w2))def=

g ((w1, w2)) if (w1, w2) V (G)g ((w1, w2)) if (w1, w2) E(G)(w12 , w

11 ) if (w1, w2) E(G1),

in V (G) E(G), for all (w1, w2) V (G) E(G). Then the map g is awell-defined graph-isomorphism. So, we can decide the groupoid-isomorphism

: G G,defined by

(W )def=

g (W ) if W V (G) E(G)g(E1) g(E2) ... g(En) if W = E1...En FPr(G) if W = ,

in G, for all W G. So, G and G are groupoid-isomorphic. !By the above two propositions, without loss of generality, we can regard

the product graphs G1 G2 and G2 G1, generated by the connected graphsG1 and G2, as the same graph, and the corresponding graph groupoids G andG are the same groupoids algebraically.

Now, let G1, ..., Gn be connected graphs, for n > 2. Then we can constructa product graph

G =nj=1

Gj .

It is easy to check that, by the previous proposition, G is graph-isomorphicto

G(i1,...,in) =nj=1

Gij ,

and hence the graph groupoid G of G is groupoid-isomorphic to the graphgroupoid G(i1, ..., in) for all (i1, ..., in) {1, ..., n}n, with ij 2= ik, wheneverj 2= k, i.e., the product graph G of G1, ..., Gn is uniquely determined up tograph-isomorphisms.


1.4.2.3 Product Graphs of Shadowed Graphs

Now, let Gk be connected graphs with their graph groupoids Gk, for k = 1,2. And let Gk be the shadowed graphs of Gk, for k = 1, 2. By considering theshadowed graphs Gk, as independent connected graphs, we can construct theproduct graph G of G1 and G2, i.e.,

Gdef= G1 G2.

Notice the difference between the shadowed graph G1 G2 of G1 G2,and the product graph G = G1 G2 of G1 and G2! As we observed in theprevious section, in general,

G1 G2Full-Subgraph

" G1 G2,and hence the graph groupoid of

G1 G2Subgroupoid

" G1 G2.Define now the unary operation (1) on V (G) E(G) by

1 : (w1, w2) % (w1, w2)1 = (w11 , w12 ),where w1k means the usual shadow of wk in V (Gk) E(Gk), for k = 1, 2.Then the graph G is self-shadowed under this operation (1), in the sensethat:

(w1, w2)1 V (G) E(G),too, whenever (w1, w2) V (G) E(G); equivalently,

(V (G) E(G))1 = V (G) E(G).This self-shadowedness of G guarantees the existence of a graph G0, whose

shadowed graph G0 is graph-isomorphic to G. Thus, we obtain the followingproposition.

Proposition 1.4.6 Let G be the product graph G1 G2 of the shadowedgraphs Gk of connected graphs Gk, for k = 1, 2. Then there exists a graphG0, such that the shadowed graph G0 is graph-isomorphic to G.

Proof. By the self-shadowedness of G, we can have a subset X0 of V (G) E(G), such that

V (G) X0,X0 X10 = V (G),

and

X0 X10 = V (G) E(G),


by the Axiom of Choice, where means the empty set, and

X10def= {W V (G) E(G) : W1 X0}.

Then we can construct a full-subgraph G0 of G by a graph with

V (G0) = V (G) X0 = V (G),and

E(G0) = E(G) X0.Then the graph G0 satisfies that the shadowed graph G0 of G0 is identical

to G. !The above proposition shows that if G is the product graph G1 G2

of the shadowed graphs Gk of connected graphs Gk, for k = 1, 2, thenthere always exists a graph G0, whose shadowed graph G0 of G0 is graph-isomorphic to G, by the self-shadowedness of G. Note that the choice ofG0 is not unique. However, the choice of such graph G0 is unique up toshadowed-graph-isomorphisms, i.e., ifG10 andG

20 satisfy the above proposition,

then the both shadowed graphs Gk0 of Gk0 , for k = 1, 2, are graph-isomorphic

to the product graph G.The important fact here is the existence of a graph G0, having its shadowed

graph G0, graph-isomorphic to the product graph G1 G2 of the shadowedgraphs Gk of Gk, for k = 1, 2.

Theorem 1.4.2 Let G = G1 G2 be the product graph of the shadowedgraphs Gk of connected graphs Gk, for k = 1, 2, and suppose G0 is a graphwhose shadowed graph G0 is graph-isomorphic to G. Then the graph groupoidG0 of G0 is groupoid-isomorphic to the product groupoid G1 G2, where Gkare the graph groupoids of Gk, for k = 1, 2.

Proof. Let G0 be the graph whose shadowed graph G0 is graph-isomorphicto the product graph G = G1 G2, and assume that it has its graph groupoidG0. Then the graph groupoidG0 ofG is generated by E(G0), since all elementsof G0 are the reduced words in E(G0). Since the shadowed graph G0 isgraph-isomorphic to the graph G, the edge set E(G0) is bijective to the edge

set E(G) of G, i.e., without loss of generality, we can identify E(G0) withE(G), i.e., the graph groupoid G0 is generated by the edge set E(G) of G,equivalently,

G0Groupoid

= the groupoid generated by E(G).

By the very definition of product groupoids, the product groupoidG = G1 G2 has its generator set E = E(G1) E(G2), i.e., all elements ofthe groupoid G are the reduced words in E . Then we have that

E(G1) E

(G2)= E

(G1 G2

)= E(G).


i.e.,

G def= the groupoid generated by EGroupoid

= the groupoid generated by E(G).

Therefore, we obtain

G0Groupoid

= the groupoid generated by E(G)Groupoid

= G,and hence, the graph groupoid G0 of G0, and the product groupoidG = G1 G2 are groupoid-isomorphic. !

The above theorem shows that the product groupoids of graph groupoidsare graph groupoids. Thus, it shows that the study of product groupoids ofgraph groupoids is the investigation of graph groupoids, too.

1.4.2.4 Certain Elements in Product-Graph Groupoids

Now, consider certain elements in the product groupoids of graph groupoids.Let Gk be the graph groupoids of connected graphs Gk, for k = 1, 2, and letG = G1 G2 be the product groupoid of G1 and G2.

We already showed that G is groupoid-isomorphic to the graph groupoidG0 of a certain graph G0, whose shadowed graphs G0 is graph-isomorphic tothe product graph G1 G2 of the shadowed graphs Gk of Gk, for k = 1, 2.

We are interested in the elements having their forms (v1, w2) or (w1, v2),

in G0, where vk V (Gk), and wk FPr(Gk), for k = 1, 2. Such elements doexist in G0 and they are contained in neither V (G0) nor E(G0), in general.

Take an element (v, e) G0, with v V (G1) and e FPr(G2). Moreover,for convenience, take e E(G2). Then we can consider this element (v, e) asa reduced word generated by the other reduced words in E(G1) E(G2), thegenerator set of G0. The interesting fact here is that (v, e) is neither a vertexnor an edge. But this element is a reduced finite path of G0!

Remark 1.4.2 By definition, the above element (v, e) is not an edge of

G0 = G1 G2. It is a reduced finite path, on G0.

Indeed, it can be understood as a reduced finite path satisfying that

(v, e) = (v, v1) (v, e) (v, v2),

whenever e = v1 e v2 E(G2), with v1, v2 V (G2), i.e., the element (v, e)can be understood as the reduced finite path having its initial vertex (v, v1)and its terminal vertex (v, v2). Similar to the case where we have (f, v) G0, where f E(G1), and v V (G2), the element (f, v) is a reduced finitepath in G0.


Example 1.4.3 Let On be the one-vertex-n-loop-edge graph, for n N.Suppose we have two graphs On1 and On2 , for n1, n2 N. Then they havetheir graph groupoids Onk , for k = 1, 2. Recall that the graph groupoidsOn of On are groups, which are group-isomorphic to the free groups Fn withn-generators, for all n N.

Consider the product groupoid O = On1 On2 . The groupoid O isgroupoid-isomorphic to the graph groupoid G0 of a graph G0, whose shadowedgraph G0 is graph-isomorphic to the product graph On1 On2 . Note that thegraph G0 has only one vertex (v1, v2), where vk are the unique vertices ofOnk , for k = 1, 2. So, the graph groupoid G0 is in fact a group (without theempty word). Also, the graph groupoid G0 has its generator set

E(On1) E(On2),with n1 n2-elements. Since every edge (e1, e2) of G0 is a loop-edge, thegraphG0 is graph-isomorphic to the one-vertex-(n1 n2)-edge graph On1n2 , andhence, our graph groupoid G0 of G0 is graph-isomorphic to On1n2 . Therefore,we obtain that

O Groupoid= G Groupoid= On1n2 Group= Fn1n2 .Therefore, we have the following isomorphism theorem for the canonical

groupoid von Neumann algebra vN(O):vN(O) -iso= MG -iso= MOn1n2

-iso= L(Fn1n2).

Example 1.4.4 Let Ge be the two-vertices-one-edge graph with V (Ge) ={v1, v2}, and E(Ge) = {e = v1ev2}. Then it has its graph groupoid Ge,identical to

{, v1, v2, e, e1},set-theoretically. Define now the product groupoid G = Ge Ge. Then Gis groupoid-isomorphic to the graph groupoid G0 of the product graph G0,whose shadowed graph G0 is graph-isomorphic to the product graph Ge Ge, where Ge is the shadowed graph of Ge. So, we can take G0 by a graphwith

E(G0) = {(e, e), (e1, e)},and

V (G0) = {(v1, v1), (v2, v2), (v1, v2), (v2, v1)}.Then the shadowed graph G0 is indeed graph-isomorphic to the product

graph Ge Ge. (Again, remark that the choice of G0 is not unique!) Thenthe graph groupoid G0 of G0 is identified with

{} V (G0) E(G0){

(v1, e), (v1, e1), (v2, e), (v2, e1),(e, v1), (e1, v1), (e, v2), (e1, v2)

},

set-theoretically.


1.4.3 Quotient Graphs

Let Gk be connected graphs with their graph groupoids Gk, for k = 1, 2.Assume that the (graph) groupoids G1 and G2 satisfy the following conditions:

(I) G2Subgroupoid G1,

(II) V (G2) = V (G1).

Then, we have the quotient groupoid G1/G2. Indeed, we can define anequivalence relation R on G1 by

(III)

w1Rw2 def w1w12 G2,for all w1, w2 G1. Then it is indeed an equivalence relation, since(i) w R w, for all w G1, by (II),(ii) if w1 R w2, then w2R w1, since w1w12 G2, if and only if

w2w11 = (w1w

12 )

1 G2,(iii) if w1 R w2, and w2 R w3, then w1 R w3, by the admissibility on

G2.

So, we can construct the equivalence classes [w] in G1, for all w G1,by

[w]def= {w G1 : w R w},

satisfying

[w] = [w], whenever w R w.

Then, on the quotient set G1/R, which is defined to be the collection ofall equivalence classes [w]s, we can define the binary operation,

(IV) [w1] [w2] = [w1w2], for all w1, w2 G1.Then the pair (G1 / R, ) is again a groupoid, and we can denote itG1/G2.In this section, we will characterize the quotient groupoid G1/G2 of the

graph groupoidG1 quotient by the graph groupoidG2, by the graph groupoidG1:2 of a certain graph G1:2.

Let G1 and G2 be connected graphs, and assume that G2 is a subpart ofG1. Recall that G2 is a part of G1, if G2 is a full-subgraph or a subgraphor a vertex subgraph of G1. Equivalently, graph K is such that K and G1are graph-isomorphic. But, for convenience, we simply understand G2 as anembedded part of G1. Denote this relation G2 is a part of G1, by

G2 G1.The relation is a partial ordering on the collection of all connected

graphs, since


(i) G G, for all connected graphs G,(ii) if G1 G2, and G2 G1, then G1 Graph= G2, and(iii) if G1 G2, and G2 G3, then G1 G3,

for connected graphs G1, G2, G3.

Suppose the given connected graphs G1 and G2 satisfy the relationG2 G1. Now, we will collapse G2 inside G1 to an ideal vertex v1:2. Forinstance, let

G2 = $$be a part of

G1 =$ $

x

.

Then we collapse G2 in G1 to a vertex v1:2. Then we obtain a new graph,

G1:2 = v1:2

x

.

More precisely, if G2 G1 are arbitrary connected graphs, then, bycollapsing G2 to a vertex v1:2, we can create a new graph G1:2, with

V (G1:2) = {v1:2} (V (G1) \ V (G2))and

E(G1:2) = E(G1) \ E(G2),with the identification rule (I R): if e = ve or e = ev in E(G1) \ E(G2), withv V (G1), and if v V (G2), then the edge e is identified with the edge inE(G1:2), also denoted by e, satisfying e = v1:2 e, respectively, e = e v1:2.

For example, in the above same example, the edge x of G1 satisfiesx = $ x , in G1. This edge x is identified with the edge x = v1:2 x inG1:2.

Definition 1.4.5 Let G1 and G2 be connected graph satisfying G2 G1.Then the ideal vertex v1:2 gotten from collapsing G2 in G1 is called the collapsedvertex of G2 in G1. And the graph G1:2 with the identification rule (I R) iscalled the quotient graph of G1 by G2. Sometimes, we denote G1:2 by G1/G2,to emphasize it is generated by the relation G2 G1.

Let G2 G1 be given connected graphs, and let G1:2 = G1/G2 be thequotient graph of G1 by G2. Then, it is an independent graph, and hence it hasits own graph groupoid G1:2. The following theorem shows that the quotientgroupoid G1/G2 is groupoid-isomorphic to the graph groupoid G1:2 of thequotient graph G1/G2, whenever the graph groupoids G1 and G2 satisfies (I)and (II).


Theorem 1.4.3 Let Gk be the connected graphs with their graph groupoidsGk, for k = 1, 2. Assume that the groupoids Gk satisfy (I), and (II), andhence we have the quotient groupoid G = G1/G2. Then the graphs G1 and G2satisfy G2 G1, and the graph groupoid G1:2 of the quotient graph G1/G2 isgroupoid-isomorphic to G.Proof. Let Gk be given as above, and assume the corresponding graphgroupoids Gk satisfy (I), and (II), for k = 1, 2. Then, the quotient groupoidG = G1/G2 is well-determined.

Since G2Subgroupoid G1, the graphsG1 andG2 satisfies the partial ordering,

G2 G1.i.e., G2 is a subpart of G1. Therefore, we can construct the quotient graphG1:2 = G1/G2 as above, and this new graph has its graph groupoid G1:2.

Assume that [w] G, i.e., [w] is an equivalence class of G1, determined bythe equivalence relation R of (III). And suppose w G2. Then this element[w] of G is identical to G2 in G1. In fact, if w G2, then

[w] = [w] = G2 in G1.Now, define the morphism

g : G G1:2by

g ([w])def=

{v1:2 if w G2 \ {}w if w (G1 \G2) {} ,

where w in the right-hand side is determined under the identification rule(I R) in G1:2. Then this morphsim g is bijective, since

|G| = |G1:2| = |(G1 \G2) {v1:2} |+ {1},where v1:2 is the collapsed vertex of G1:2, and

{1} def={

1 if G0 if / G.

Moreover, this map g satisfies

g ([w1][w2]) = (g ([w1])) (g ([w2])) ,

in G1:2, for all [w1], [w2] G. Therefore, the bijective morphism g is agroupoid-isomorphism. Equivalently, the quotient groupoid G = G1/G2 isgroupoid-isomorphic to the graph groupoid G1:2 of the quotient graphG1:2 = G1/G2. !

The above theorem shows that the quotient groupoid G1/G2 of the graphgroupoid G1 by the graph groupoid G2 is groupoid-isomorphic to the graphgroupoid G1:2 of the quotient graph G1/G2 of G1 by G2. So, the study ofquotient groupoids of graph groupoids (satisfying (II)) is investigating certaingraph groupoids. The above theorem is independently proven in [8], but ourproof here is simple and new.


Remark 1.4.3 Without the conditions (I) and (II) on graph groupoids, forany arbitrary subpart inclusion G2 G1, we can freely construct thecorresponding quotient graph G1/G2, i.e., the construction of G1/G2 is notdependent upon any conditions or any assumptions, for G2 G1. Thereforethe corresponding graph groupoid G1:2 of the quotient graph G1/G2 is naturallydetermined.

However, the construction of the quotient groupoid G1/G2 of G1 and G2 isrestricted, i.e. we need the conditions (I) and (II) (or we need certain pull-backprocesses instead of (II)), i.e., the construction of G1/G2 is dependent uponthe restricted conditions.

The above theorem shows that, if the quotient groupoid G1/G2 exists,then it is groupoid-isomorphic to the graph groupoid G1:2 of the quotient graphG1/G2.

In summary, if G1/G2 is a quotient graph with its graph groupoid G1:2,then G1:2 is not groupoid-isomorphic to the quotient groupoid G1/G2, wheneverthe graph groupoids G1 and G2 satisfy neither (I) nor (II) (or a certainpull-back process), since G1/G2 is not constructed. But if the quotient groupoidG1/G2 is well-determined, then it is groupoid-isomorphic to the graph groupoidG1:2 of the quotient graph G1/G2.

The following example is interesting, since it provides a quotient relationon free groups.

Example 1.4.5 Let Onk be the one-vertex-nk-loop-edge graph with its graphgroupoid Onk , for k = 1, 2. We observed that the graph groupoids Onk aregroups, which are group-isomorphic to the free groups Fnk with nk-generators,for k = 1, 2. Assume now that n2 < n1 in N. Then

On2 On1 ,

and hence

On2Subgroupoid On1 .

The groupoids On1 and On2 satisfy that: V (On1) = V (On2). Therefore,we can obtain the quotient groupoid O1:2 = On1/On2 .

Consider the quotient graph O1:2 = On1/On2 . It is easy to check that thisgraph O1:2 is graph-isomorphic to the graph On1n2 . So, the graph groupoidO1:2 of O1:2 is groupoid-isomorphic to the graph groupoid On1n2 of theone-vertex-(n1 n2)-loop-edge graph On1n2 .

Therefore, we can obtain that:

O1:2Groupoid

= O1:2Groupoid

= On1n2Group= Fn1n2 .


Lets consider the following fundamental example.

Example 1.4.6 Let G2 be a graph,

G2 =

,

which is a full-subgraph of G1,

G1 =

e

.

Then the graph groupoids Gk of Gk, for k = 1, 2, satisfy the conditions(I) and (II). Thus, the quotient groupoid X = G1/G2 is well-determined.

Consider now the quotient graph G1:2 = G1/G2 of G1 by G2. By thecollapsing process, we obtain that

G1:2 =

v1:2!e,

where v1:2 is the collapsed vertex of G2 in G1, and e of G1:2 is the edgegotten from e of G1, under the identification rule (I R). Therefore, the graphgroupoid G1:2 of G1:2 is groupoid-isomorphic to the graph groupoid O1 ofthe one-vertex-1-loop-edge graph O1. Recall that the groupoid O1 is a group,which is group-isomorphic to the infinite abelian cyclic group Z. Therefore,

X Groupoid= G1:2 Groupoid= O1 Group= Z.From the above two examples, we can get the following refined version of

the above theorem.

Theorem 1.4.4 Let G1 and G2 be graph groupoids of connected graphs G1and G2, respectively, satisfying (I) and (II). Let G1/G2 be the quotientgroupoid of G1 by G2. Then there exists N N {} , such that G1/G2is group-isomorphic to the free group FN .

Proof. The condition (II) for G1 and G2 is crucial here. Since the basesV (Gk) of Gk are identical, the quotient graph G1/G2 of G1 by G2 has onlyone vertex, which is the collapsed vertex v1:2 of G2 in G1. Therefore, under(II),

V (G1/G2) = {v1:2}.Now, let

N = |E(G1 / G2)| N {} .Indeed, since

|E(G1 / G2)| = |E(G1) \ E(G2)|= |E(G1)| |E(G2)| ,

the quantity N is contained in N {} .Therefore, the quotient graph G1/G2 is graph-isomorphic to the one-

vertex-N -loop-edge graph ON . Thus, the graph groupoid G1:2 of G1/G2 isgroupoid-isomorphic to the graph groupoid ON of ON . So,


G1/G2Groupoid

= G1:2Groupoid

= ONGroup= FN .

!

The above theorem provides the very nice tool to characterize the quotientgroupoids generated by graph groupoids, and the graph groupoids of quotientgraphs. It is very interesting that, under the connectedness, the quotientgroupoids of graph groupoids (with (II)) are group-isomorphic to free groups.

1.4.4 Complemented Graphs

As before, let Gk be connected graphs with their graph groupoids Gk, fork = 1, 2. It is convenient to assume that both graphs G1 and G2 are subpartsof a sufficiently big graph K. Then, like set-substraction, we can define a newgraph G = G1 G2, which is another part of K, by a graph with the edgeset,

E(G) = E(G1) \ E(G2)and its vertex set

V (G) = {v, v : e E(G) s.t., e = vev}.For instance, let

G1 = "

", and G2 =

" "

,

where the graphs G1 and G2 share their common subpart

"".

Thus, by definition, we can obtain

G1 G2 = "

",

and

G2 G1 ="

.

Here, we provide the easy steps to construct G1 G2.(1) In G1, get rid of all edges of G2(2) Consider the remaining edges. If there are vertices which are not connected

by remaining edges, then get rid of them, too.


(3) The resulting graph is the graph G1 G2.Notice here that it is possible the graphs G1 and G2 are disjoint, in the

sense that:

E(G1) E(G2) = = V (G1) V (G2).In such a case, we define

G1 G2 def= G1, and G2 G1 def= G2.In fact, by following the above construction, we can get

G1 G2 = G1, and G2 G1 = G2,whenever G1 and G2 are disjoint. But, just for sure, lets determine the abovespecial cases, as definitions.

Definition 1.4.6 Let G1 and G2 be connected graphs. The new graphG1 G2 is called the complemented graph of G2 in G1. Define the subset1:2 of the vertex set V (G1 G2) and V (G2) by the intersection,

1:2def= V (G1 G2) V (G2).

This subset 1:2 of V (G1 G2) is called the boundary of G2 in G1.Then we can obtain the following theorem.

Theorem 1.4.5 Let Gk be connected graphs, for k = 1, 2, and letG = G1 G2 be the complemented graph of G2 in G1, with its boundary1:2. If 1:2 2= , then

G1Graph= (G1 G2) G2,

where means the union on graphs.Indeed, if 1:2 = , then

(G1 G2) G2 = G1 unionsq G2Graph

2= G1.!

Thanks to the above theorem, we can obtain that:

Theorem 1.4.6 Let Gk be connected graphs with their graph groupoids Gk,for k = 1, 2, and let G = G1 G2 be the complemented graph of G2 in G1.Assume that the boundary 1:2 is nonempty. If G is the graph groupoid of G,then

G1Groupoid

= G + G2.!

1.5 Bibliography 41

Theorem 1.4.7 Let Gk be connected graphs with their graph groupoids Gk,for k = 1, 2, and let G = G1 G2 be the complemented graph of G2 in G1.Assume that the boundary 1:2 is nonempty. Then the graph groupoid G of Gis groupoid-isomorphic to

G Groupoid= G1 G2,where G1 G2 is the complemented groupoid of G2 in G1. !

The above theorem shows that the complemented groupoids of graphgroupoids are again characterized by other graph groupoids, in particular,the graph groupoids of complemented graphs.

Summary

In this chapter, we considered algebras induced by directed graphs. Everydirected graph G induces its natural algebraic structure G, the graph groupoidof G. Moreover, by our main results, every operated groupoid X of graphgroupoids G1 and G2 is groupoid-isomorphic to the graph groupoid G ofa certain operated graph of two graphs G1 and G2, generating G1 and G2,respectively. Therefore, the study of operated graph groupoids is theinvestigation of graph groupoids of operated graphs. !

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I. Cho, Distorted Histories, (2009) Published by Verlag with Dr. Muller.

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Algebras, Graphs and Their Applications