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Chapter 9

- Graphs

Sec 9.1

- Graphs and Graph Models

Simple Graph

- A simple graph G is an ordered pair (V,E),

consisting of a nonempty set of vertices V, and a

set of edges E, where each edge eE corresponding

to a unique element in u,v uV, vV, u v

- For an edge e which corresponds to u,v the

vertices u and v are called the endpoints of this

edge

Multigraph

- A multigraph G is an ordered pair (V,E),

consisting of a nonempty set of vertices V, and a

set of edges E, where each edge e E corresponds

to some element in u,v uV, vV, u v - Note It is possible for two edges to correspond

to the same u,v , allowing for two or more

edges to have the same endpoints.

Pseudograph

- A pseudograph G is an ordered pair (V,E),

consisting of a set of vertices V, and a set of

edges E, where each edge e E corresponds to

some element in u,v uV, vV u,u

uV - Note that an edge may correspond to u thus

allowing both endpoints to be the same. Such

edges are called loops.

Directed Graph

- A directed graph G is an ordered pair (V,E),

consisting of a set of vertices V, and a set of

directed edges E, where each edge e E

corresponds to an element in (u,v) uV, vV - Note Associating directed edges to ordered

pairs means that each edge has an associated

orientation or direction. If e corresponds to

(u,v), then u is called the initial vertex and v

is called the terminal vertex of e.

Simple Directed Graph

- A directed graph G is simple if it has no loops

and no multiple directed edges. So a simple

directed graph is an ordered pair (V,E),

consisting of a set of vertices V, and a set of

edges E, where each edge e E corresponds to at

a unique element in (u,v) uV, vV, u v

Directed Multigraph

- A directed multigraph G is an ordered pair (V,E),

consisting of a set of vertices V, and a set of

edges E, where each edge e E corresponds to

some element in (u,v) uV, vV - Note More than one edge may correspond to the

same (u,v) allowing multiple directed edges

between two vertices.

Summary Graph G (V,E)

- Simple Each edge ei a unique u,v, u v
- Multigraph Each edge ei some u,v, u v
- Pseudograph Each edge ei some u,v or u
- Simple Directed Graph Each edge ei a unique

(u,v), u v - Directed Multigraph Each edge ei some (u,v)

Homework

- Sec 9.1
- pg. 596 1, 3, 5, 7, 9, 15, 21

Sec 9.2

- Graph Terminology

Definitions

- Adjacent Vertices Two vertices, u and v in an

undirected graph G are adjacent (or neighbors) if

there is an edge e in E which corresponds to

u,v - Incident If edge e corresponds to u,v, the

edge e is called incident with the vertices u and

v. - Connected If edge e corresponds to u,v, the

vertices u and v are said to be connected. - Endpoints If edge e corresponds to u,v, the

vertices u and v are said to be endpoints of e

Degree of a Vertex

- Deg(v) The degree of a vertex in an undirected

graph G is the number of edges incident with it.

(with the convention that a loop at v contributes

twice to the degree of that vertex) - The Handshaking Theorem. Let G (V,E) be an

undirected graph. The sum of the degrees of all

the vertices of the graph is equal to 2E. - Theorem An undirected graph has an even number

of vertices of odd degree.

Definition Directed Graphs

- Let G is a directed graph with an edge e

corresponding to (u,v), - Adjacent Then vertex u is said to be adjacent to

v and vertex v is said to be adjacent from u. - Initial and Terminal Vertices Vertex u is called

the initial vertex and vertex v is called the

terminal vertex of edge e. - For a loop in a directed graph, the single vertex

is considered both the initial and the terminal

vertex.

In-degree and Out-degree

- Deg-(v) In a directed graph G, the in-degree of

vertex v is the number of edges with v as their

terminal vertex. - Deg(v) In a directed graph G, the out-degree

of vertex v is the number of edges with v as

their initial vertex. - Theorem For any directed graph G (V,E), If

the summation is taken over all v V, deg(v)

deg-(v) E2e, e of edges

Some Special Simple Graphs

- Complete A simple graph G is complete if it

contains an edge between each pair of vertices.

A complete graph of n vertices is denoted by Kn. - Bipartite A simple graph G (V,E) is called

bipartite if the vertices in V can be partitioned

into two disjoint sets V1 and V2 so that each

edge e E has exactly one endpoint in V1 and the

other in V2.

Results

- Theorem A simple graph is bipartite if and only

if it is possible to assign one of two different

colors to each vertex of the graph so that no two

adjacent vertices are assigned the same color. - Complete Bipartite A bipartite graph whose

vertices can partitioned into two sets of m and n

vertices respectively with an edge between every

pair of vertices in the two sets is called a

complete bipartite graph, denoted by Km,n.

Application Job Assignments

- Suppose that there are m employees in a group and

j different jobs that need to be done where m

j. Each employee is trained to do one or more of

there j jobs. - Vertices
- Employees
- Jobs
- Edges connect employees to jobs

Homework

- Sec 9.2
- pg. 608 1, 3, 5, 7, 9, 17, 21, 23

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