5 Graph

5) Key edge: edge e={ vi，vj} be erased from G, then { the components of G increase,{ vi，vj} is called key edge. Exe 10 {v4，v5} and{ v4，v6} are both key key edge， edge，
Chart 5 Graph
Section 1 Fundamental Conception
Definition 5-1 A graph G consists of a finite set V of
objects called vertices and a finite set E called edges which is a binary relation on V, denoted by G(V,E). Exe. 1
Exe4
=2， =2，
the following graph 6-25(a) mines u W(G {u}) W(G–{u})
（b）W（G–{ u1，u2}）=3. {

Theorem 6-8 Let G be a graph with n vertices, if the sum of any two vertices of G no less than n–1, 1 then G has a H-path. path. Theorem 5-9 Let G be a graph with more than 3
7. Degree of a vertex: the number of edges having that vertex as an end point. 8. Loop: the special edge that ends the same vertex. 9. Regular graph: that with all the vertices have the same degree denoted as Dn.
Exe.3
三、Connected graph
Theorem 5-1 Let G(V,E) be a graph with V={v1, v2, … ,vn } and |E|=m, then
For any graph G, then number of even. vertices with the odd degree must be even. Proof Let graph G have set V1 consisting vertices with odd degree, V2 consisting vertices with even degree, and m edges. then so
Lamar
as
and 2m be both even, so
even. Be even. For v∈V1 ， deg(v) is odd, then|V1| even. be even.
9．Path: the sequence of edges{v0，v1}{v1，v2}…{vl–1，vl} ． {v 1 with one adjacent the next is called a path connect v0 to vl with the length of l. or simple denoted by sequence of vertices v0v1v2…vl–1vl. 10． path： 10．Open path：if v0≠vl, then v0v1v2…vl–1vl is a open path. v 1 11． 11．Circuit: if v0=vl，v0v1v2…vl–1vl is circuit. v 1 12． path： 12．Simple path：open path v0v1v2…vl–1vl，with no vertex v 1 repeated.or called real path. 13．cycle： 13．cycle：simple path with the same ending vertex. 14． vertices： 14．Connected of two vertices：there is a path between them.
5Theorem 5-7
Let G=（V，E）be a H-graph, then G=（ H-
V， W(G–S) for any nonempty set S of V，there is W(G S) ≤|S| W(G–S) where W(G S) denotes the number of components when G mines S |S| is the number of vertices in S.
3) Component Let G(V,E) be a graph and disconnected, the connected pieces are called the components of the graph G(V,E).
4)Key vertex: if vertex v be erased from G (including all the edges ends it), then the components increase. v is called key Vertex.
theorem 5-4
Exe 2 :
Section 2 Euler Graph and Hamiltonian Graph
• 一 Euler Graph • 1 Definition Let G(V,E) be a connected graph, if there is a path including every edge exactly once, the path is called Euler path; if the Euler path is a circuit then path is called Euler circuit; G(V,E) is called Euler graph if it has a Euler circuit.
2. Distance: the number of edges of shortest path between vi and vj is the distance of vi and vj denoted by d(vi, vj).
Theorem 5-3 Let G(V,E) be a graph with |V|=n, then any real path in G has the length no more than n-1. Proof:
E = 设 V ={v1，v2，v3，v4，v5},
{v1, v2}, {v1, v3}, {v2, v3}, {v2, v4}, {v3, v4},
graph. then G=(V,E) is a graph. 。 (a).(b) we use pictures to represent it as.
四、sub graph and component Definition 5-5 Let G1=（V1,,E1） G2=（V2，E2）be graphs.
（1）if V2 denoted （2）if V2 ⊆ V1, E2 ⊆ E1, then G2 is sub graph of G1. by G2 ⊆ G1; = V1, E2 ⊆ E1, then G2 is generated by G1.
Theorem 6-6 The connected graph G has a Euler path if
and only if there are two vertices has odd degree.
Exe.2
N
Y
Y
N
二、Hamiltonian Graph
Definition 6-12 If graph G has a real/simple circuit exactly contains every vertex only once, then the graph is called Hamiltonian graph and the circuit is called Hamiltonian circuit; if the simple path contains every vertex only once then the path is Hamiltonian path.
六） Matrix description Definition 5-8 Let G=(V,E) be a graph where V={v1，v2，…，vn}, then square matrix of A=(aij) ， of n is called adjacent matrix of G, the term of row_i and collum_j is defined as:
Exe which of the following graph is euelr graph.
Theorem 5-5 The connected graph G is Euler graph if
and only if the degree of any vertex in G is even.
二、Complete graph
1．G(n，m): G(n，m): 2．vi adjacent vj: 3．e=(vi,vj) ends with vi, vj 4．Isolated vertex edges: 5．Adjacent edges: 6．Isolated edge Def. Def.5-2 Let G(V,E) be a graph, if any two vertices are adjacent, then G is called complete graph. Exe. graph. Exe.3 1-complete, 2-complete … 5-complete graph
theorem 5-2 { vi, vj } is key edge if and only if { vi, vj } is not an edge of any real\ real circuit of G.
五, shortest path and distance
1．Shortest path: within a graph G, vi connects ． to vj by more than one path, the shortest one is called shortest path.
5Definition 5-4
In a graph G, if any two
vertices are connected by a path, then G is a connected graph, else G is disconnected. Graph with only one isolated vertex is also considered connected graph. Exe.
vertices, if the sum of any two vertices of G no less than n, then G is a H-graph. graph.
Exe.
H-path
H-circuit
Section 3 Undirected Trees
Exe 1
图6-21
Let G be a graph with V={v1， v2 ， … ， vn} and adjacent matrix A, then the ) term aij(l) of Al （ l=1,2,… ） is the number of = path from vito vjwith the length l.