G-v, or G-{v}

1. Definition 8: Graphs that have a number assigned to each edge or each vertex are called weighted graphs • weighted digraphs

2. Definition 9: The graph G'(V',E') is called a subgraph of G(V,E) If V'V and E'E. If V'=V, then G'(V,E') is said to be a spanning subgraph.

3. Definition 10: If G'(V',E') contains all edges of G that join two vertices in V' then G' is called the induced subgraph by V'V and is denoted by G(V'). • induced subgraph by {v1,v2,v4,v5}

5. 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

6. 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

7. 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

9. 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.

10. 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

11. (e6,e7,e8,e4,e7) is not a circuit; (e1,e6,e7,e8,e4,e5) is a circuit (e1,e8,e4,e5) 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 fromv2to v1 • (e8,e4,e5) is a simple path of from v2to v1

12. 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

13. 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.

14. components of the graph G1,G2,…,Gω

15. 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

16. 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

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

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

19. 5.2.3 Connectivity in directed graphs • Definition 16: Let n be a nonnegative integer and G be a directed graph. A path of length n from u to v in G is a sequence of edges e1,e2,…,enof 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. 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 v0,v1,…,vnare all distinct.

20. (e1,e2,e7,e1,e2,e7)is not a circuit (e1,e2,e7,e6,e12) is a circuit (e1,e2,e7) is a simple circuit. (a,b,c,a) • (e1,e2,e7,e1,e2,e9)is not a path • (e1,e2,e7,e6,e9)is a path from ato e • (e1,e2,e9)is a path from ato e, is a simple path. • (a,b,c,e)

21. Definition 17: A directed graph is strongly connected if there is a path from a to b and from b to a whenever a and b are vertices in the graph. A directed graph is connected directed graph if there is a path from a to b or b to a whenever a and b are vertices in the graph. A directed graph is weakly connected if there is a path between every pair vertices in the underlying undirected graph.

22. (a)strongly connected • (b)connected directed • (c)weakly connected • strongly connected components: G1,G2,…,Gω

23. V ={v1,v2,v3,v4,v5,v6,v7, v8} • V1={v1,v7,v8}, V2={v2,v3,v5,v6}, V3={v4}, • strongly connected components : • G(V1),G(V2),G(V3)

24. Next: Bipartite graph, Euler paths and circuits, P296 8.2

25. Exercise P128 11;P295 11,17,19,22,23,28 • 1.Prove that the complement of a disconnected graph is connected. • 2.Let G be a simple graph with n vertices. Show that ifδ(G) >[n/2]-1, then G is connected. • 3.Show that a simple graph G with n vertices are connected if it has more than (n-1)(n-2)/2 edges. • 4.Represent each of these graphs with an adjacency matrix an incidence matrix.