Chapter 1 Chapter 1 Fundamental Concepts IIFundamental Concepts IIPao-Lien Lai
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DefinitionsCountingThe pigeonhole principleGraphic sequencesDegrees and digraphs
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DefinitionsDefinitions
degree of v : ◦number of non-loop edges containing v plus twice the number
of loops containing v.
(G) : (\Delta) maximum degree of G.(G) : (\delta) minimum degree of G.k-regular : (G) = (G) = k .
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DefinitionsDefinitions
Isolated vertex : degree=0.Neighborhood : NG(v) , NG[v]n(G), |G| :
◦order of G , is the number of vertices in G.e(G) : the number of edges in G.
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CountingCounting5
(Degree Sum Formula) If G is a graph with vertex degree d1,…,dn,
then the summation of all di = 2e(G).
)()(2)(
GVvGevd
CountingCounting
In a graph G, the average vertex degree is , and hence
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)(
)(2
Gn
Ge
)()(
)(2)( G
Gn
GeG
Every graph has an even number of vertices of odd degree.
No graph of odd order is regular with odd degree.
A k-regular graph with n vertices has nk/2 edges.
Example Example
k-dimensional cube (hypercube Qk)Vertices: k-tuples with entries in {0,1} Edges: the pairs of k-tuples that differ in exactly one position. j-dimensional subcube: a subgraph isomorphic to Qj.
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Q3
ExampleExample
Structure of hypercubes◦Parity of vertex: the number of 1s◦Two independent sets
Each edge of Qk has an even vertex and an odd vertex. Bipartite graph
◦k-regular◦n(Qk)=2k. e(Qk)=k2k-1.
◦Two subgraphs of Q3 are isomorphic to Q2.
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The Graph MenagerieThe Graph Menagerie 動物園動物園
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triangle claw 爪 paw 爪子 kite 鳶
Petersen graphPetersen graph
The simple graph whoseVertices:
◦2-element subsets of 5-element setEdges :
◦the pairs of disjoint 2-element subsets
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12
3445
23 51
3552
2441
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The pigeonhole principleThe pigeonhole principle13
(Pigeonhole Principle) If a set consisting of more than kn objects is partitioned into n classes, then some class receives more than k objects.
Theorem1:Every simple graph with at least two vertices has two vertices of equal degree.
{0,1,……,n-1} 0 and n-1 both occurs impossibly
The pigeonhole principleThe pigeonhole principle14
Theorem 2:If G is a simple graph of n vertices with (G) (n-1)/2, then G is connected.
Example Example
Let G be the n-vertex graph with components isomorphic to and .
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2/nK 2/nK
2/nK 2/nK
12/)( nG
G is disconnected
* Induction trap* Induction trap
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Every 3-regular simple connected graph has no cut-edge.
False conclusion!!
CounterexampleCut edge
Degree sequenceDegree sequence17
degree sequence : the list of vertex degrees, in nonincreasing order, d1…dn.
Proposition Proposition
The nonnegative integers d1, d2, …, dn are the vertex degrees of some graph if and only if is even.
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n
i id1
Graphic sequencesGraphic sequences19
graphic sequence : a list of nonnegative numbers that is the degree sequence of some simple graph
ExampleExample
A recursive condition
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The lists 1,0,1 and 2,2,1,1 are graphic
The list 2,0,0 is not graphic
ExampleExample21
The list 33333221 is graphic
33333221w2223221
3222221v111221
221111u10111
11110
The realization is not unique!
u
v
u
v
u
w
Graphic sequencesGraphic sequences22
Graphic Theorem:For n > 1, the nonnegative integer list d of size n is graphic if and only if d’ is graphic, where d’ is the list of size n-1 obtained from d by deleting its largest and subtracting 1 from its next largest elements. (The only 1-element graphic sequence is d1=0)
DigraphsDigraphs23
A directed graph or digraph G is a triple consisting of a vertex set V(G), and edge set E(G), and a function assigning each edge an ordered pair of vertices
Tail: the first vertex of the ordered pairHead: the second vertex of the ordered pairEndpoints: tail and headAn edge: from tail to head tail head
DigraphsDigraphs
Loop: an edge whose endpoints are equalMultiple edges:
◦edges having the same ordered pair of endpoints.Simple graph:
◦each ordered pair is the head and tail of at most one edge◦One loop may be present at each vertex
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DigraphsDigraphs
In a simple graph◦An edge uv: tail u and head vFrom u to v
◦v is a successor of u◦u is a predecessor of v
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u v
ApplicationApplication
Finite state machine
Markov chain
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DD- UD+
DU+ UU-
DD+ UD-
DU- UU+
G B
.2
.3
.7
.8
DigraphsDigraphs
Path◦A simple digraph whose vertices can be linearly ordered so
that there is an edge with tail u and head v if and only if v immediately follows u in the vertex ordering
Cycle◦Defined similarly using an ordering of the vertices on a
circuit.
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ExampleExample
Functional digraph of f◦The simple digraph with vertex set A and edge set
{(x,f(x):xA)}◦For each x, the single edge with tail x points to the image of
x under f.Permutation
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7
1
2
4
3 5
6
001010010100100001111111
011110110101101011
DigraphsDigraphs
Underlying graph 相關圖 of a digraph D◦The graph G obtained by treating the edges of D as
unordered pairs◦The vertex set and edge set remain the same◦The endpoints of an edge are the same in G as in D◦But the edge become an unordered pair in G.
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D G
Example Example 30
ab
cdx
y z
ab
cdx
w
y z
0100
1021
0201
0110
z
y
x
w
zyxw
10000
11110
01101
00011
z
y
x
w
edcba
A(G) M(G)
0000
1010
0101
0100
z
y
x
w
zyxw
A(D)
10000
11110
01101
00011
z
y
x
w
edcba
M(D)
DigraphsDigraphs
Weakly connected◦Underlying graph is connected
Strongly connected (strong)◦For each ordered pair u,v of vertices, there is a path from u
to v.Strong components
◦Maximal strong subgraphs
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Example Example 32
x y
a b c d e
a b c d e
Not strongly connected
5 strong components
1 strong component
3 strong components
Degrees and digraphsDegrees and digraphs33
Out-degree : d+(v) v is tail. (out-neighborhood N+(v) )
In-degree : d-(v) v is head. (in-neighborhood N-(v) )
Minimum in-degree: -(G)Maximum in-degree:Δ-(G)Minimum out-degree: +(G)Maximum out-degree: Δ+(G)
PropositionProposition
In a digraph G,
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)()()()()(
vdGevdGVvGVv
Eulerian DigraphsEulerian Digraphs
Eulerian trail◦A trail containing all edges
Eulerian circuit◦A closed trail containing all edges
Eulerian◦A digraph is Eulerian if it has an Eulerian circuit
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LemmaLemma
If G is a digraph with +(G)1, then G contains a cycle. The same conclusion holds when -(G)1.
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uMaximal path Pv u
Theorem Theorem
A digraph is Eulerian if and only if d+(v)=d-(v) for each vertex v and the underlying graph has at most one nontrivial component.
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ApplicationApplication
De Bruijn cycles◦2n binary strings of length n◦Is there a cyclic arrangement of 2n binary digits such
that the 2n strings of n consecutive digits are all distinct?For example:
◦n=4◦0000111101100101 works
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00000001001101111111111011011011
0
1
00
0
11
1011
0
0
01
1 01101100100100100101101001001000
Example Example 39
1o
ooo
o
o
1
1 1 11
1
o
o
001
000
011
010
100
1
110
111101
D4
TheoremTheorem
The digraph Dn is Eulerian, and the edge labels on the edges in any Eulerian circuit of Dn from a cyclic arrangement in which the 2n consecutive segments of length n are distinct.
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ExampleExample 41
00000001001101111111111011011011
0
1
00
0
11
1011
0
0
01
1 01101100100100100101101001001000
1o
ooo
o
o
1
1 1 11
1
o
o
001
000
011
010
100
1
110
1111010
12
3 4
56
7 8
9
10
11 12
1314
15
01234567
89101112131415
Degrees and digraphsDegrees and digraphs42
An orientation of graph G: a digraph D obtained from G by choosing an
orientation (xy or yx) for each edge xyE(G).
An orientation graph is an orientation of a simple graph
tournament 比賽 : complete graph and each edge with orientation.
ExampleExample
Consider an n-team league where each team plays every other exactly once.◦For each pair u,v
Include the edge uv if u wins Include the edge vu if v wins
At the end◦There is an orientation of Kn
◦The score of a team is its outdegree
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Exercise 1.3.8Exercise 1.3.8
Which of the following are graphic sequences? Provide a construction or a proof of impossibility
for each◦(5,5,4,3,2,2,2,1)◦(5,5,4,4,2,2,1,1)◦(5,5,5,3,2,2,1,1)◦(5,5,5,4,2,1,1,1)
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Exercise 1.4.19 or 1.4.20Exercise 1.4.19 or 1.4.20
A digraph is Eulerian if and only if d+(v)=d-(v) for each vertex v and the underlying graph has at most one nontrivial component.
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