Discrete_Math_(Combinatorics).pdf - Discrete Math 1 Graph Theory 1.1 Introduction to graphs Basic definitions and properties of graphs Simple

# Discrete_Math_(Combinatorics).pdf - Discrete Math 1 Graph...

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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 .  #### You've reached the end of your free preview.

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