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.

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Model of Computer Networks

Model of Computer Networks

• Graphs that may have multiple edges connecting the same vertices are called multigraphs.

• multiplicity of an edge

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• 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.

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• undirected graphs• undirected edges• simple directed graphs• directed multigraphs (multiple directed edges,

multiplicity)• mixed graphs

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Simple directed graphs

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Directed multigraphs

• Niche Overlap Graphs in Ecology

Graph Models

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• Acquaintanceship Graphs

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• Influence graphs

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• The Hollywood graph

• Round Robin Tournament

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• Call graphs• The Web Graph• Roadmaps

• Precedence Graphs and Concurrent Processing

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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

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• 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=610. Thus e=30

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

• Proof. Let V1 and V2 be the set of vertices of odd degree and the set of even degree.

Thus, |V1| is even.

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• 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.

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

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