G-v, or G-{v} When we remove a vertex v from a graph, we must remove all edges incident with the vertex v. When a edge is removed from a graph, without removing endpoints of the edge

Adjacency matrices and Incidence matrices Definition 12: Let G(V,E) be a graph of non-multiple edge where |V|=n. Suppose that v1,v2,…,vn are the vertices. The adjacency matrix A of G, with respect to this listing of the vertices, is the nn zero-one matrix with 1 as its (i,j)th entry when vi and vj are adjacent, and 0 as its (i,j)th entry when they are not adjacent. In other words, If its adjacency matrix is A=[aij], then

Let G(V,E) be an undirected graph Let G(V,E) be an undirected graph. Suppose that v1,v2,…,vn are the vertices and e1,e2,…,em are the edges of G. Then the incidence matrix with respect to this ordering of V and E is the nm matrix M=[mij], where

Quotient graph Definition 13: Suppose G(V,E) is a graph and R is a equivalence relation on the set V. We construct the quotient graph GR in the follow way. The vertices of GR are the equivalence classes of V produced by R. If [v] and [w] are the equivalence classes of vertices v and w of G, then there is an edge in GR between [v] and [w] if some vertex in [v] is connected to some vertex in [w] in the graph G.

5.2 Paths and Circuits 5.2.1 Paths and Circuits Definition 14: Let n be a nonnegative integer and G be an undirected graph. A path of length n from u to v in G is a sequence of edges e1,e2,…,en of G such that e1={v0=u,v1}, e2={v1,v2},…,en={vn-1,vn=v}, and no edge occurs more than once in the edge sequence. When G is a simple graph, we denote this path by its vertex sequence u=v0,v1,…,vn=v. A path is called simple if no vertex appear more than once. A circuit is a path that begins and ends with the same vertex. A circuit is simple if the vertices v1,v2,…,vn-1 are all distinct

(e6,e7,e1) is a path of from v2 to v1 (e6,e7,e8,e4,e7) is not a circuit; (e1,e6,e7,e8,e4,e5) is a circuit (e1,e5,e4,e8) is a simple circuit (e6,e7) is a simple circuit (e6,e7,e8,e4,e7,e1) is not a path; (e6,e7,e1) is a path of from v2 to v1 (e8,e4,e5) is a simple path of from v2 to v1

Theorem 5.4:Let  (G)≥2, then there is a simple circuit in the graph G. Proof: If graph G contains loops or multiple edges, then there is a simple circuit. (a,a) or (e,e'). Let G be a simple graph. For any vertex v0 of G, d(v0)≥2, next vertex, adjacent, Pigeonhole principle

5.2.2 Connectivity Definition 15: A graph is called connectivity if there is a path between every pair of distinct vertices of the graph. Otherwise , the graph is disconnected.

components of the graph G1,G2,…,Gω

A graph that is not connected is the union of two or more connected subgraphs, each pair of which has no vertex in common. These disjoint connected subgraphs are called the connected components of the graph

Example: Let G be a simple graph Example: Let G be a simple graph. If G has n vertices, e edges, and ω connected components , then Proof: e≥n-ω Let us apply induction on the number of edges of G. e=0, isolated vertex，has n components ，n=ω, 0=e≥n-ω=0，the result holds Suppose that result holds for e=e0-1 e=e0, Omitting any edge , G', (1)G' has n vertices, ω components and e0-1 edges. (2)G' has n vertices, ω+1 components and e0-1 edges

2. Let G1,G2,…,Gωbe ω components of G. Gi has ni vertices for i=1,2,…, ω, and n1+n2+…+nω=n，and

If G is connected, then the number of edges of G has at least n-1 edges. Tree.

Next: Connectivity in directed graphs P135 4. 3 (Sixth) OR 4 Next: Connectivity in directed graphs P135 4.3 (Sixth) OR 4.3(Fifth), Bipartite graph, Euler paths and circuits, P311 8.2 (Sixth) OR P296 8.2(Fifth) 期中测验时间：11月4日

Exercise P310 (Sixth) OR P295(Fifth) 11, 17,22,28 1.Represent each of these graphs with an adjacency matrix an incidence matrix.