1 Discrete Math 1 Graph Theory 1.1 Introduction to graphs • Basic definitions and properties of graphs. • Simple mathematical models of networks. • Concepts like Isomorphism as well as special types, such as connected graphs and bipartite graphs are discussed. 1.1.1 Graph Models • Always look out for things or objects and relations between pairs of objects when modeling graphs: 1. Things or objects will become vertices in the graphs; 2. Relations between pairs of objects will become edges of the graph. • A graph G(V,E) consists of a finite set V of vertices and a set E of edges joining different pairs of distinct vertices. • Vertices a and b are adjacent when there is an edge (a,b). • Directed graphs contain ordered pairs of vertices, called directed edges. • A directed edge is denoted by ( g1854 g4652g1318 ,c) to show the direction from b to c. • A path P is a sequence of distinct vertices, written P=x 1 -x 2 -...-x n -x 1 , which each pair of consecutive vertices in P jointed by an edge. (p.1 notes) • If there is an edge (x n ,x 1 ), the sequence is called a circuit , written as x 1 -x 2 -...- x n -x 1 . (p.1 notes) • A graph is connected if there is a path between every pair of vertices. • The removal of certain edges or vertices from a connected graph is said to disconnect the graph and the resulting graph is no longer connected. • If at least one pair of vertices is no longer joined by a path, then the graph is disconnected. • A graph in which all edges go horizontally between two sets of vertices is called bipartite . • The number of edges incident to a vertex is called the degree of the vertex . • A set C of vertices in a graph with the property that every edge is incident to at least one vertex in C is called an edge cover . • A set of vertices without an edge between any two is called an independent set of vertices .
2 1.1.2 Isomorphism • Two graphs G and G' are called isomorphic if there exists a one-to-one correspondence between the vertices in G and the vertices in G' such that a pair of vertices are adjacent in G if and only if the corresponding pair of vertices are adjacent in G'. • To be isomorphic, two graphs must have the same number of vertices and the same number of edges . • Degrees of vertices are preserved under isomorphism, that is, two matched vertices must have the same degree . • Two isomorphic graphs must have the same number of vertices of a given degree . • A subgraph G' of a graph G is a graph formed by a subset of vertices and edges of G. • If two subgraphs are isomorphic , then subgraphs formed by corresponding vertices and edges must be isomorphic . • A graph with n vertices in which each vertex is adjacent to all the other vertices is called a complete graph on n vertices, denoted K n .