1. Chapter 5 Graphs the puzzle of the seven bridge in the Königsberg,
on the Pregel

3. Kirchhoff
Cayler CnH2n+1
The four colour problem四色问题
Hamiltonian circuits
1920s,König: finite and infinite graphs
OS,Compiler,AI, Network

4. 5.1 Introduction to Graphs
5.1.1 Graph terminology
Relation: digraph
Definition 1 : Let V is not empty set. A directed graph, or digraph, is an ordered pair of sets (V,E) such that E is a subset of the set of ordered pairs of V. We denote by G(V,E) the digraph. The elements of V are called vertices or simply "points", and V is called the set of vertices. Similarly, elements of E are called "edge", and E is called the set of edges.

5. E={(a,b),(a,c),(b,c),(c,a),(c,c),(c,e),(d,a),(d,c),(f,e), (f,f)},
Edge (a,b)
a: initial vertex，
b:terminal vertex
edges (a,b) incident with the vertices a and b。
(c,c),(f,f) loop
g: isolated vertex。
G=(V,E),V={a,b,c,d,e,f,g},
E={(a,b),(a,c),(b,c),(c,a),(c,c),(c,e),(d,a),(d,c),(f,e), (f,f)},

6. Definition 2：Let (a,b) be edge in G
Definition 2：Let (a,b) be edge in G. The vertices a and b are called endvertices of edges; a and b are called adjacent in G; the vertex a is called initial vertex of edge (a,b), and the vertex b is called terminal vertex of this edge. The edge (a,b) is called incident with the vertices a and b. The edge (a,a) is called loop。The vertex is called isolated vertex if a vertex is not adjacent to any vertex.
g is an isolated vertex, (c,c) ,(f,f) are loop. a and b are adjacent; c and d are adjacent;

7. Definition 3: Let V is not empty set
Definition 3: Let V is not empty set. An undirected graph is an ordered pair of sets (V,E) such that E is a sub-multiset of the multiset of unordered pairs of V. We denote by G(V,E) the graph. The elements of V are called vertices or simply "points", and V is called the set of vertices. Similarly, elements of E are called "edge", and E is called the set of edges.
V={v1,v2,v3,v4,v5,v6}，E={{v1,v2},{v1,v5,}，{v2,v2}, {v2,v3},{v2,v4},{v2,v5},{v2,v5},{v3,v4},{v4,v5}}，
edges {v1,v2} incidents with the vertices v1 and v2
loop ; isolated vertex
edge {v2,v5} multiple edge。

8. Definition 4：These edges are called multiple edges if they incident with the same two vertices. The graph is called multigraph. The graph is called a simple graph, if any two vertices in the graph, may connect at most one edge (i.e., one edge or no edge) and the graph has no loop. The complete graph on n vertices, denoted by Kn, is the simple graph that contains exactly one edge between each pair of distinct vertices.

9. undirected graph: graph
finite graph
finite digraph

10. Definition 5：The degree of a vertex v in an undirected graph is the number of edges incident with it, except that a loop at a vertex contributes twice to the degree of that vertex. The degree of the vertex v is denoted by d(v). A vertex is pendent if only if it has degree one. The minimum degree of the vertices of a graph G is denoted by  (G)(=minvV{d(v)}) and the maximum degree by  (G)(=maxvV{d(v)}
b=a,{a,a},

12. Theorem 5.2: An undirected graph has an even number of vertices of odd degree.

13. Definition 6：In a directed graph the out-degree of a vertex v by d+(v) is the number of edges with v as their initial vertex. The in-degree of a vertex v by d-(v), is the number of edges with v as their terminal vertex. Note that a loop at a vertex contributes 1 to both the out-degree and the in-degree of this vertex. The degree of the vertex v is denoted by d(v).

14. Theorem 5.3: Let G(V,E) be an directed graph. Then

15. aD, bB,cA,dE;
(a,b)(D,B), (a,c)(D,A),…，
isomorphism

16. Definition 7：The directed graphs G(V,E) and G'(V',E') be isomorphic if there is a one to one and onto everywhere function f from V to V' with the property that (a, b) is an edge of G if only if (f(a),f(b)) is an edge of G'. We denote by GG'. The undirected graph G(V,E) and G'(V',E') be isomorphic if there is a one to one and onto everywhere function f from V to V' with the property that {a, b} is an edge of G if only if {f(a),f(b)} is a edge of G'. We denote by GG'.

18. Petersen
3-regular
The graph is called k-regular if every vertex of G has degree k.

19. Next: Paths and Circuits, Connectivity,8.1 P291
Exercise P123 27,28; P ,10;