9 Graphs

A graph G = (V, E) consists of V, a nonempty set of vertices (or nodes) and E, a set of edges. Each edge has either one or two vertices associated with it, called its endpoints. An edge is said to connect its endpoints. A graph with an infinite vertex set is called an infinite graph, and a graph with finite vertex is called finite graph. In this book, we usually consider only finite graphs.

A graph in which each edge connects two different vertices and where no two edges connect the same pair of vertices is called a simple graph. San Francisco Denver Chicago Washington Detroit New York Los Angeles Model of Computer Networks

Graphs that may have multiple edges connecting the same vertices are called multigraphs. multiplicity of an edge San Francisco Denver Chicago Washington Detroit New York Los Angeles

Sometimes a communication link connects a data center to itself for diagnosis purpose. Such edges are called loops. Graphs that includes loops and possibly multiedges are called pseudographs. San Francisco Denver Chicago Washington Detroit New York Los Angeles

undirected graphs undirected edges simple directed graphs directed multigraphs (multiple directed edges, multiplicity) mixed graphs

Denver Chicago Washington Detroit New York Los Angeles Simple directed graphs

Denver Chicago Washington Detroit New York Los Angeles Directed multigraphs

Niche Overlap Graphs in Ecology Graph Models Shrew Opossum WoodpeckerMouse Owl Squirrel Hawk Raccoon Crow

Acquaintanceship Graphs A B GF E H C D I M M K L J

Influence graphs B H C D I M M K

The Hollywood graph

Round Robin Tournament B H D I M C

Call graphs The Web Graph Roadmaps

Precedence Graphs and Concurrent Processing S1S1 S4S4 S3S3 S5S5 S6S6 S2S2

9.2 Graph Terminology and Special Types of Graphs Two vertices u and v in an undirected graph G are called adjacent (or neighbors) in G if u and v are endpoints of an edge of G. If e is associated with {u,v}, the edge e is called incident with the vertices u and v. The edge e is also said connect u and v. The vertices u and v are called endpoints of an edge associated with {u,v}.

The degree of a vertex 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 a vertex. The degree of the vertex v is denoted by deg(v). Example: What are the degrees of the vertices in the graphs G and H displayed below. G H a a b bc c d d e e f g

Theorem: (The Handshaking Theorem) Let G = (u,v) be an undirected graph with e edges. Then Example: How many edges are there in a graph with 10 vertices each of degree six? 2e=6  10. Thus e=30

Theorem: An undirected graph has an even number of vertices of odd degree. Proof. Let V 1 and V 2 be the set of vertices of odd degree and the set of even degree. Thus, |V 1 | is even.

When (u,v) is an edge of the graph G with directed edges, u is said to be adjacent to v and v is said to be adjacent from u. The vertex u is called the initial vertex of (u,v), and v is called the terminal or end vertex of (u,v). The initial vertex and the terminal vertex of a loop are the same.

In a graph with directed edges the in-degree of a vertex v, denoted by deg - (v), is the number of edges with v as their terminal vertex. The out-degree of a vertex v, denoted by deg + (v), is the number of edges with v as their initial vertex. Example: Find the in-degree and out-degree of each vertex of the following graph. ab c d e f

Theorem: Let G = (V,E) be a graph with directed edges. Then