chapter 1 fundamental concepts ii pao-lien lai 1
TRANSCRIPT
![Page 1: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/1.jpg)
Chapter 1 Chapter 1 Fundamental Concepts IIFundamental Concepts IIPao-Lien Lai
1
![Page 2: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/2.jpg)
DefinitionsCountingThe pigeonhole principleGraphic sequencesDegrees and digraphs
2
![Page 3: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/3.jpg)
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 .
3
![Page 4: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/4.jpg)
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.
4
![Page 5: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/5.jpg)
CountingCounting5
(Degree Sum Formula) If G is a graph with vertex degree d1,…,dn,
then the summation of all di = 2e(G).
)()(2)(
GVvGevd
![Page 6: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/6.jpg)
CountingCounting
In a graph G, the average vertex degree is , and hence
6
)(
)(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.
![Page 7: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/7.jpg)
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.
7
Q3
![Page 8: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/8.jpg)
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.
8
![Page 9: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/9.jpg)
The Graph MenagerieThe Graph Menagerie 動物園動物園
10
triangle claw 爪 paw 爪子 kite 鳶
![Page 10: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/10.jpg)
Petersen graphPetersen graph
The simple graph whoseVertices:
◦2-element subsets of 5-element setEdges :
◦the pairs of disjoint 2-element subsets
11
12
3445
23 51
3552
2441
13
![Page 11: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/11.jpg)
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
![Page 12: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/12.jpg)
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.
![Page 13: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/13.jpg)
Example Example
Let G be the n-vertex graph with components isomorphic to and .
15
2/nK 2/nK
2/nK 2/nK
12/)( nG
G is disconnected
![Page 14: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/14.jpg)
* Induction trap* Induction trap
16
Every 3-regular simple connected graph has no cut-edge.
False conclusion!!
CounterexampleCut edge
![Page 15: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/15.jpg)
Degree sequenceDegree sequence17
degree sequence : the list of vertex degrees, in nonincreasing order, d1…dn.
![Page 16: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/16.jpg)
Proposition Proposition
The nonnegative integers d1, d2, …, dn are the vertex degrees of some graph if and only if is even.
18
n
i id1
![Page 17: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/17.jpg)
Graphic sequencesGraphic sequences19
graphic sequence : a list of nonnegative numbers that is the degree sequence of some simple graph
![Page 18: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/18.jpg)
ExampleExample
A recursive condition
20
The lists 1,0,1 and 2,2,1,1 are graphic
The list 2,0,0 is not graphic
![Page 19: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/19.jpg)
ExampleExample21
The list 33333221 is graphic
33333221w2223221
3222221v111221
221111u10111
11110
The realization is not unique!
u
v
u
v
u
w
![Page 20: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/20.jpg)
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)
![Page 21: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/21.jpg)
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
![Page 22: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/22.jpg)
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
24
![Page 23: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/23.jpg)
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
25
u v
![Page 24: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/24.jpg)
ApplicationApplication
Finite state machine
Markov chain
26
DD- UD+
DU+ UU-
DD+ UD-
DU- UU+
G B
.2
.3
.7
.8
![Page 25: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/25.jpg)
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.
27
![Page 26: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/26.jpg)
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
28
7
1
2
4
3 5
6
001010010100100001111111
011110110101101011
![Page 27: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/27.jpg)
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.
29
D G
![Page 28: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/28.jpg)
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)
![Page 29: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/29.jpg)
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
31
![Page 30: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/30.jpg)
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
![Page 31: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/31.jpg)
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)
![Page 32: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/32.jpg)
PropositionProposition
In a digraph G,
34
)()()()()(
vdGevdGVvGVv
![Page 33: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/33.jpg)
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
35
![Page 34: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/34.jpg)
LemmaLemma
If G is a digraph with +(G)1, then G contains a cycle. The same conclusion holds when -(G)1.
36
uMaximal path Pv u
![Page 35: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/35.jpg)
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.
37
![Page 36: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/36.jpg)
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
38
00000001001101111111111011011011
0
1
00
0
11
1011
0
0
01
1 01101100100100100101101001001000
![Page 37: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/37.jpg)
Example Example 39
1o
ooo
o
o
1
1 1 11
1
o
o
001
000
011
010
100
1
110
111101
D4
![Page 38: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/38.jpg)
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.
40
![Page 39: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/39.jpg)
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
![Page 40: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/40.jpg)
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.
![Page 41: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/41.jpg)
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
43
![Page 42: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/42.jpg)
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)
44
![Page 43: Chapter 1 Fundamental Concepts II Pao-Lien Lai 1](https://reader036.vdocuments.us/reader036/viewer/2022062423/56649ec75503460f94bd2ed5/html5/thumbnails/43.jpg)
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.
45