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

 Kirchhoff  Cayler C n H 2n+1  The four colour problem 四色问题  Hamiltonian circuits  1920s,König: finite and infinite graphs  OS,Compiler,AI, Network

5.1 Introduction to Graphs  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.

 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)}, 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 。

 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;

 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={v 1,v 2,v 3,v 4,v 5,v 6 } ， E={{v 1,v 2 },{v 1,v 5,} ， {v 2,v 2 }, {v 2,v 3 },{v 2,v 4 },{v 2,v 5 },{v 2,v 5 },{v 3,v 4 },{v 4,v 5 }} ， edges {v 1,v 2 } incidents with the vertices v 1 and v 2 loop ; isolated vertex edge {v 2,v 5 } multiple edge 。

 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 K n, is the simple graph that contains exactly one edge between each pair of distinct vertices.

 undirected graph: graph  finite graph  finite digraph

 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)(=min v  V {d(v)}) and the maximum degree by  (G)(=max v  V {d(v)}  b=a,{a,a},

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

 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).

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

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

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

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

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

 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.

 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 {v 1,v 2,v 4,v 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

Adjacency matrices and Incidence matrices  Definition 12: Let G(V,E) be a graph of non- multiple edge where |V|=n. Suppose that v 1,v 2,…,v n 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 v i and v j are adjacent, and 0 as its (i,j)th entry when they are not adjacent. In other words, If its adjacency matrix is A=[a ij ], then

 Let G(V,E) be an undirected graph. Suppose that v 1,v 2,…,v n are the vertices and e 1,e 2,…,e m are the edges of G. Then the incidence matrix with respect to this ordering of V and E is the n  m matrix M=[m ij ], where

 Paths and Circuits, Connectivity,8.1 P291

 Exercise P123 27,28;  P295 9,10;  1.Represent each of these graphs with an adjacency matrix an incidence matrix.