LATTICE THEORETIC APPROACH TO THE STUDY OFIRREGULAR GRAPHS

Selvam Avadayappan1 & P. Santhi2

Let G = (V, E) be a finite simple connected graph. For any vertex v in V, the set of vertices adjacent to v be the neighbourhoodof v, and is denoted by N(v). A connected graph G is called highly irregular if every vertex of G is adjacent only to verticeshaving distinct degrees. H

n,n denotes the highly irregular bipartite graph with bipartite sets X = {v

1, v

2, ..., v

n} and Y = {u

1, u

2,

…, un} and with edge set E = { v

iu

j/ 1 ≤ i ≤ n, n – i + 1 ≤ j ≤ n}. A connected graph G is said to be neighbourly irregular (or

simply NI) if any two adjacent vertices have distinct degrees. A special type of NI graph which contains at least one vertexof each possible degree is denoted by I

n. (I

ndenote the graph with vertex set {v

1, v

2, ..., v

n} and edge set

{ , 1 , – }1– 2 ≤ ≤ ≤ ≤ +

nv v i i j n ijn i which is NI if and only if n is odd.) In this paper we apply the lattice

theoretic approach to the irregular graphs In and H

n, nand we classify these graphs as Boolean graphs and ortho graphs.

Keywords: Regular Graphs, Highly Irregular Graphs, Neighbourly Irregular Graphs, Ortho Graphs, Boolean Graphs.

AMS Subject Classification Code: 05C (primary)

1. INTRODUCTION

Throughout this paper, we consider only finite simpleconnected graphs. For notations and terminology we follow[3]. It is known that there is not a graph with vertices ofdistinct degrees, and any non – trivial graph must have atleast two vertices of the same degree. Behzad – Chatrand[4] and Nebesky [10] have studied the graphs containingexactly two vertices of the same degree. We name this graphas I

n graph [2], a graph which contains at least one vertex

of each possible degree. Ebrahim Salehi called this as apairlone graph [11] and proved that, for any n, there existsa unique pairlone graph of order n.

Let In denote the graph with vertex set {v

1, v

2, ..., v

n}

and edge set , 1 , -1- 2

≤ ≤ ≤ ≤ +

nv v i i j n ijn i .

Degree of each vertex vi,

( ),

2

-1,2

ni if i

d vi ni if i

≤ = >

For example the graphs and are shown in Figure 1.

1Department of Mathematics, V.H.N.S.N. College, Virudhunagar-626 001

2Department of Mathematics,C.K.N. College for Women,Cuddalore-607 001

selvam_avadayappan@yahoo.co.in, santhi_amuthu@yahoo.com

Fig. 1

A connected graph G is called highly irregular if everyvertex of G is adjacent only to vertices with distinct degrees[1]. That is, if two vertices u and v of G are both adjacent toa vertex w of G, then d(u) ≠ d(v) in G. For example, thegraphs given in Figure 2 are highly irregular.

Fig. 2

Hn,n

denotes the highly irregular bipartite graph [12] withbipartite sets X = {v

1, v

2, ...,v

n} and Y = {u

1, u

2, …, u

n} and

with edge set E = {viu

j/ 1 ≤ i ≤ n, n – i + 1 ≤ j ≤ n}. In, H

n,n,

d(ui) = d(v

i) = i for 1 ≤ i ≤ n. For example, the graph shown

in Figure 3 is H5,5

.

G. Birkhoff [5] introduced a lattice theoretic approachto the study of graphs. In lattice theory, we have ortho lattices,orthomodular lattices, modular ortholattices and Boolean

International Journal of Information Technology and Knowledge ManagementJuly-December 2010, Volume 2, No. 2, pp. 317-322

318 SELVAM AVADAYAPPAN & P. SANTHI

Algebras. According to these lattices E.K.R. Nagarajan andM.S. Mutharasu [8], [9] classify the graphs as ortho graphs,orthomodular graphs, modular orthographs and Booleangraphs. In [5], G.Birhoff has described a method of assigningan ortholattice for each graph.

Fig. 3

An ortholattice is a bounded lattice L equipped with afunction c : L → L satisfying

(i) x ≤ y ⇒ c(y) ≤ c(x),

(ii) c(c(x)) = x, and

(iii) x ≤ c(x) ⇒ x = 0

for all x and y in L. (See [5] for the basic terminology oflattices and partially ordered sets. In particular, note that 0is often used to denote the least element of the lattice.) Afunction satisfying condition (i) is called antitone. Thefunction c is called the orthocomplementation of theortholattice. A complete ortholattice is an ortholattice whichis a complete lattice.

Given a graph G, we will construct a completeortholattice L(G), which we call the neighbourhoodortholattice of G. (The construction was described byBirkhoff [5] and studied in [13].)

Let 2G denote the lattice of all subsets of the vertex setof G. Define a function γ : 2G → 2G by γ(A) = {x ∈ V(G) / xis adjacent to ‘a’ for all a ∈ A}. In other words, γ(A) is theset of all common neighbours of A. Clearly γ is antitoneand satisfies γ(γ(A)) γ ⊇ A for every subset A of G. It followsthat γ(γ(γ(A))) = γ(A). A subset A of V(G) is said to be closedif γ(γ(A)) = A.

The set all closed subsets of V(G) is a complete latticeL(G) when ordered by inclusion and the meet operation isordinary set intersection. The map γ restricts to anorthocomplementation of L(G). (Property (iii) of anorthocomplementation follows from the fact that no vertexis adjacent to itself.) Two subsets A and B of vertices of Gare orthogonal if and only if every vertex of A is adjacentto every vertex of B. Also, for any subset A of V(G), γ(A) =∩{γ({a}) / γ ∈ A} where γ({a}) is the neighbourhood of a.Therefore we obtain an alternate description of theunderlying lattice of L(G): take the set of neighbourhoods,and close with respect to arbitrary intersections. We regardthe null intersection as giving the entire set G of vertices,

which is the largest element of L(G). For example considerthe graph given in Figure 4.

Fig. 4

Here γ({a}) = {b, c} and γ(γ({a})) = γ({b, c}) = {a}.Hence {a} ∈ L(G). Similarly {b}, {c} ∈ L(G). Nowγ(γ({d})) = γ({c, e}) = {b, d} ≠ {d} and γ(γ({e})) = γ({b,d}) = {c, e} ≠ {e}. Therefore {d}, {e} ∉ L(G).

Consider γ(γ({a, b})) = γ({c}) = {a, b, e} ≠ {a, b} andγ(γ({b, c})) = γ({a}) = {b, c} and so {b, c} ∈ L(G) and{a, b} ∉ L(G). In a similar way we can prove that {b, e},{c, d}, {a, c, d}, {a, b, e} ∈ L(G) and the remaining vertexsubsets are not in L(G). Thus L(G) = {φ, V(G), {a}, {b},{c}, {b, c}, {b, e}, {c, d}, {a, c, d}, {a, b, e}}. The Hassediagram of the underlying lattice of the neighbourhoodortholattice is shown in Figure 5.

Fig. 5

A graph G is said to be an orthomodular graph if thelattice of G, namely L(G), is an orthomodular lattice. A graphG is said to be a modular orthograph if the lattice of G is amodular ortholattice. A graph G is a Boolean graph if thelattice of G is a Boolean Algebra. As every graph gives riseto an ortholattice, we call every graph as an orthograph [8].It has been proved [7] that an ortholattice is orthomodularif and only if it has no isomorphic copy of O

6, where O

6is

the lattice given in Figure 6.

LATTICE THEORETIC APPROACH TO THE STUDY OF IRREGULAR GRAPHS 319

Fig. 6

2. LATTICE OF IRREGULAR GRAPHS

In this chapter, we apply the lattice theoretic approach tothe irregular graphs I

n and H

n, n and we classify these graphs

as Boolean graphs and orthographs respectively.

First, we consider the graph In.

Consider the graph I2 shown in Figure 7.

Fig. 7

Here γ({v1})= {v

2} and γ(γ({v

1})) = γ({v

2}) = {v

1}.

Also, γ(γ({v2}) = γ({v

1}) = {v

2}. Thus {v

1}, {v

2} ∈ L(I

2).

Since there exists no vertex in I2 which is adjacent to two or

more other vertices of I2, there exists no three or more

element vertex subset L(I2) in except V(I

2). Hence L(I

2) =

{φ, V(I2), {v

1}, {v

2}} and the corresponding lattice diagram

is given in Figure 8.

Fig. 8

This implies that L(I2) is isomorphic to B

2, Boolean

algebra of 2 atoms. Therefore, the graph I2 is a Boolean

graph.

Next we consider the graph shown in Figure 9.

For the graph I3, γ({v

1})} = {v

3} and γ(γ(v

1})) = γ({v

3})

= {v1, v

2} ≠ {v

1} and so, {v

1} ∉ L(I

3). Similarly{v

2} ∉ L(I

3).

(since, 2( ({ })vγ γ = 3({ })vγ = 1 2{ , }v v ≠ {v2}.) But, 3( ({ }))vγ γ

= 1 2({ , })v vγ = {v3}. Hence {v

3} ∈ L(I

3). Also,

1 2( ({ }))v vγ γ =

Fig. 9

3({ })vγ = 1 2{ , }v v and 1 3( ({ , }))v vγ γ = ( )γ φ =V(I3)≠ 1 3{ , }v v .

Hence {v1, v

2} ∈ L(I

3) and 1 3{ , }v v ∉ L(I

3). Similarly,,

2 3{ , }v v ∉ L(I3). There exists no three or more element

vertex subset in L(I3) except V(I

3) since no vertex in I

3which

is adjacent to three or more other vertices of I3. Hence L(I

3)

= {φ, V(I3), {v

1, v

2}, {v

3}}. The lattice diagram L(I

3) is given

in Figure 10. This diagram L(I3) is isomorphic to B

2and

hence I3is a Boolean graph.

Note that L(I2) ≅ L(I

3).

Fig. 10

Similarly, for the graph I4 shown in Figure 11, N(v

1) =

{v4}, N(v

2) = {v

3, v

4}, N(v

3) = {v

2, v

4} and N(v

4) = {v

1, v

2,v

3}

and L(I4) = {φ, V(I

4), {v

2}, {v

3}, {v

4}, {v

2, v

4}, {v

3, v

4}, {v

1,

v2, v

3}}. The corresponding lattice diagram is given in Figure

12 which is isomorphic to B3-Boolean algebra of 3 atoms

and hence the graph I4 is a Boolean graph.

Fig. 11

320 SELVAM AVADAYAPPAN & P. SANTHI

Fig. 12

Also, for the graph I5 shown in Figure 13, N(v

1) = {v

5},

N(v2) = {v

4, v

5}, N(v

3) = {v

4, v

5}, N(v

4) = {v

2, v

3, v

5} and

N(v5) = {v

1, v

2, v

3, v

4}, the lattice of I

5, L(I

5) = {φ, V(I

5),

{v4}, {v

5}, {v

2, v

3}, {v

4, v

5}, {v

2, v

3, v

5}{v

1, v

2, v

3, v

4}}, and

the corresponding lattice diagram is shown in Figure 14.

Fig. 13

Hence L(I4) ≅ L(I

5) ≅ 3.

From the above discussion we have conclude thefollowing theorem:

Fig. 14

Theorem 5.3.1: Lattice of In is isomorphic to 1

2

Bn +

,

Boolean Algebra of 12

n + atoms and so the graph I

n is a

Boolean graph.

Proof: For the graph I2n

, the elements of vertex subsetsof ( )2nL I are given below:

( )2nL I = {φ, {vn}, {v

n+1}, …, {v

2n}, {v

n, v

n+2}, {v

n, v

n+3},

…,{vn, v

2n}, {v

n+1, v

n+2}, …, {v

n+1, v

2n}, …, {v

2n–1, v

2n}, {v

n–1,

vn, v

n+1}, {v

n, v

n+2, v

n+3}, …, {v

2n–2, v

2n–1, v

2n}, {v

n–1, v

n, v

n+1,

vn+3

}, …, {v2n–3

, v2n–2

, v2n–1

, v2n

}, {vn–2

, vn–1

, vn, v

n+1, v

n+2},

…,{v2, v

3,... , v

2n–2}, …, {v

2, v

3,... , v

2n–2,v

2n}, {v

1, v

2,... , v

2n–2,

v2n–1

}}. The atoms of 2( )n

L I are {vn}, {v

n+1}, …, {v

2n}.

For the graph I2n+1

,

( )2 1L I n+ = {φ, {vn+2

}, {vn+3

}, …, {v2n

}, {v2n+1

}, {vn,

vn+1

}, {vn+2

, vn+3

}, …, {vn+2

, v2n+1

}, {vn+3

, vn+4

}, …, {vn+3

, v2n+1

},

…, {v2n

, vn+1

}, {vn, v

n+1, v

n+3}, {v

n, v

n+1, v

n+4}, …, {v

2n–1, v

2n,

v2n+1

}, {vn–1

, vn, v

n+1, v

n+2}, …,{v

2n–2, v

2n–1, v

2n, v

2n+1}, {v

n–1, v

n,

vn+1

, vn+2

,vn+4

}, …, {v2, v

3,...,v

2n–1}, …, {v

2, v

3,...,v

2n–1, v

2n+1},

{v1, v

2,...,v

2n–1, v

2n}}. The atoms of L(I

2n+1) are {v

n+2}, {v

n+3},

…, {v2n

}, {v2n+1

}, {vn, v

n+1}. Thus L(I

2n) and L(I

2n+1) contains

2n+1 elements and consequently isomorphic to Booleanalgebra of n + 1 atoms.

Thus L(In) is isomorphic to 1

2

Bn +

, the Boolean

algebra of 12

n + atoms and hence I

n is a Boolean graph.

Next, we will discuss the lattice of the highly irregularbipartite graph H

n,n.

Note that H1,1

is nothing but P2whose lattice L(P

2) is

isomorphic to B2. This means that H

1,1is a Boolean graph.

Theorem 5.3.2: Highly irregular bipartite graph, Hn,n

,≥ 2 is an ortho graph. That is, the lattice of H

n,n is an ortho

lattice.

Proof: First we prove this result for the cases whenn = 2 and 3.

Consider H2,2

shown in Figure 15. In H2,2

, N(u1) = {v

2},

N(u2) = {v

1, v

2}, N(v

1) = {u

2} and N(v

2) = {u

1, u

2}.

Now γ(γ({u1}) = γ({v

2}) = {u

1, u

2} ≠ {u

1}, and γ(γ({u

2}))

= γ({v1, v

2}) = {u

2}. Hence {u

1} ∉ L(H

2,2) and {u

2} ∈ L(H

2,2).

Similarly {v1} ∉ L(H

2,2) and {v

2} ∈ L(H

2,2). Also, γ(γ({v

1,

v2})) = γ({u

2}) = {v

1, v

2} and γ(γ({u

1, u

2})) = γ({v

2}) = {u

1,

LATTICE THEORETIC APPROACH TO THE STUDY OF IRREGULAR GRAPHS 321

u2}. Consequently {u

1, u

2}, {v

1, v

2} ∈ L{H

2,2}. Thus L(H

2,2)

= {φ, V(H2,2

), {v1, v

2}, {u

1, u

2}, {v

2}, {u

2}}. Corresponding

lattice diagram is shown in Figure 16.

Fig. 15

L(H2,2

) ≅ 6. If a lattice contains 6, then it is anortholattice, consequently H

2,2is an ortho graph.

For the graph H3,3

shown in Figure 17, L(H3,3

) = {φ,V(H

3,3), {v

1, v

2, v

3}, {u

1, u

2, u

3}, {v

2, v

3}, {u

2, u

3}, {v

3}, {u

3}},

since γ(γ({v3})) = γ({u

1, u

2, u

3}) = {v

3}, γ(γ({v

2, v

3})) = γ({u

2,

u3}) = {v

2, v

3}, γ(γ({v

1, v

2, v

3})) = γ({u

3}) = {v

1, v

2, v

3}.

Similarly {u3}, {u

2, u

3}, {u

1, u

2, u

3} ∈ L(H

3,3).

Fig. 16

Fig. 17

Corresponding lattice diagram is given in Figure 18.

Fig. 18

Now we prove the general case. For a graph Hn,n

, N(v1)

= {un}, N(v

2) = {u

n–1, u

n}, ...., N(v

i) = {u

n(i–1), ..., u

n–1, u

n},

N(vn) = {u

1, u

2, ..., u

n–1, u

n} and N(u

1) = {v

n}, N(u

i) = {v

n–(i–1),

..., vn–1

, vn} and the lattice of H

n,n, L(H

n,n) = {φ, V(H

n,n), {v

1,

v2..., v

n}, {u

1, u

2, ..., u

n–1, u

n}, {v

2, v

3..., v

n}, {u

2, u

3, ..., u

n–1,

un}, ..., {v

n}, {u

n}}. The corresponding lattice diagram of

L(Hn,n

) is given in Figure 19.

The lattice L(Hn,n

) contains a sub lattice which isisomorphic to O

6 and is given by the elements {φ, {v

n}, {u

n},

{v1, v

2..., v

n}, {u

1, u

2, ..., u

n–1, u

n}, V(H

n,n)} and hence H

n,nis

an ortho graph. Hence the highly irregular bipartite graphsH

n,n, n ≥ 2 are ortho graphs.

Fig. 19

322 SELVAM AVADAYAPPAN & P. SANTHI

ACKNOWLEDGEMENT

One of the authors P. Santhi would like to thank theUniversity Grants Commissions, New Delhi, for theirsupport (Grant No. XTFTNMD 134 / FIP – X PLAN) forthis research work, and V.H.N.S.N. College, Virudhunagar,for their hospitality.

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