Graph Theory

Graphs are discrete structures consisting of vertices and edges that connects these vertices. There are several types of graphs that differ with respect to the kind and number of edges that can connect a pair of vertices.

A simple graph G = (V, E) consists of V, a nonempty set of vertices, and E, a set of unordered pairs of distinct elements V called edges.

A multigraph G = (V, E) consists of a set V of vertices, a set of E of edges, and a function f from E to {{u, v} | u, v є V, u ≠ v}. The edges e1 and e2 are called multiple or parallel edges if f(e1) = f(e2).

A pseudograph G = (V, E) consists of a set V of vertices, a set E of edges, and a function f from E to {{u, v} | u, v є V}. An edge is loop of f(e) = {u, u} = {u} for some u є V.

A directed graph (V, E) consists of a set of vertices V and a set of edges E that are ordered pairs of elements V.

A directed multigraph G = (V, E) consists of a set of vertices V and a set of edges E, and d function f from E to {(u, v) | u, v ε V}. The edges e1 and e2 are multiple edges if f(e1) = f(e2).

Summary

Graph Terminology

Type Edges Multiple edges

Loops

Simple Graph Undirected No No

Multigraph Undirected Yes No

Pseudograph Undirected Yes Yes

Directed Graph Directed No Yes

Directed multigraph Directed Yes Yes

Basic Terminology

Two vertices u and v in an undirected graph G are called adjacent (or neighbors) in G if {u, v} is an edge of G. If e = {u, v}, the edge e is called incident with the vertices u and v. The edge e is also said to connect u and v. The vertices u and v are called endpoints of the edge {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 that vertex. The degree of the vertex v is denoted by deg(v).

A vertex of degree 0 is called isolated.A vertex is pendant if and only if it has degree 1.

Example: What are the degrees of the vertices in graphs G and H.

G

a

b c d

gef

H

a b c

de

Theorem 1

The Handshaking Theorem Let G = (V, E) be an undirected graph with e edges. Then

veVv

deg2

Example: How many edges are there in a graph with 10 vertices each of degree 6.

2e = (10)(6)2e = 60e = 30

Theorem 2

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

When (u, v) is an edge of the graph G with directed edges, u is said to adjacent to u 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 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 v, denoted by deg + (v), is the number of edges with v as their initial vertex. (Note that a loop at a vertex contributes 1 to both the in-degree and the out-degree of this vertex.)

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

EvvVvVv

degdeg

Example: Find the in-degree and out-degree of each vertex in the graph shown below with directed edges.

a b

c

de

f

Cycles – the cycle Cn, n ≥ 3, consists of n vertices v1, v2,…., vn and edges {v1, v2}, {v2, v3},…, {vn-1, vn}, and {vn, v1}.

C3 C4 C5 C6

Wheels – we obtain the wheel Wn when we add an additional vertex to the cycle Cn, for n ≥ 3, and connect this new vertex to each of the n vertices Cn, by new edges.

W3 W4 W5 W6

Representing GraphsAdjacency lists – is used to represent a graph with no multiple edges which specify the vertices that are adjacent to each vertex of the graph.

Vertex Adjacent Vertices

A B, C, E

B A

C A, D, E

D C, E

E A, C, D

A

B

C

E D

Vertex Adjacent Vertices

A B, C, D, E

B B, D

C A, C, E

D

E B, C, D

A

B

C

E D

Adjacency matrix (Ag), with respect to this listing of the vertices, is the n x n zero-one matrix with 1 as its (i, j)th entry when vi and vj are adjacent, and 0 as its (i, j) entry when they are not adjacent. In other words, if its adjacency matrix is A = [aij], then

0

1ija

If {vi, vj} is an edge of G,

otherwise.

a b

c d

0001

0011

0101

1110

0110

1001

1001

0110a b

d c

Adjacency matrices can also be used to represent undirected graphs with loops and with multiple edges. A loop at the vertex ai is represented by 1 at the (i, j)th position of the adjacency matrix. When multiple edges are present, the adjacency matrix is no longer zero-one matrix, since the (i, j) entry of this matrix equals the number of edges that are associated to {ai, aj}. All undirected graphs, including multigraphs and pseudographs, have symmetric adjacency matrices

0212

2100

1003

2030a b

d c

Incidence matrices – Let G = (V, E) be an undirected graph. Suppose that v1, v2, …, vn are the vertices and e1, e2, …., em are the edges of G. Then the incidence matrix with respect to this ordering of V and E is the n x m matrix M = [mij], where

0

1ijm

when edge ej is incident with vi,

otherwise.

v1 v2 v3

v4 v5

e1

e2

e3

e4 e5

e6

011010

000101

110000

101100

000011

Exercises: I. Represent the following using adjacency and incidence matrix.

1. a b

c d

e1

e2

e3e4 e5

2.a b

d c

e1e2

e3e4 e5

e6e7

e8

a b

d c

e1 e2 e3 e4

e5

e6

e7

e8

e9 e10

e11 e12

3.

II. Draw the graph represented by the given adjacency matrix.

111

100

101

.4

120

202

021

.5

0101

1130

0302

1021

.6

III. Draw the graph and represent it using adjacency matrix.

7. K4 8. K6 9. C1010. W6

Shortest Path Problems

Applications: mileage, flight times, fares, distance, response time, least rates, and so on.

Weighted graphs are graphs that have a number assigned to each edge.

The length of a path in a weighted graph is the sum of the weights of the edges of this path.

Theorem 2

Dijktra’s algorithm uses O(n2) operations (additions and comparisons) to find the length of the shortest path between two vertices in a connected simple undirected weighted graph.

Dijktra’s Algorithm (a shortest path algorithm)

Theorem 1

Dijktra’s algorithm finds the length of a shortest path between two vertices in a connected simple undirected weighted graph.

Find the length of the shortest path between a and z in the given weighted graph.

a

b

c

d

e

f

g

z

4

3

2

5

3

6

1

5

5

2

7

4

The traveling salesman problem

It asks for the circuit of minimum total weigh in a weighted, complete, undirected graph that visits each vertex exactly once and returns to its starting point.

Detroit

Toledo

Saginaw

Ground Rapids

Kalamazoo133

58

98

113

56

147

137 142

135

167

Route Total distance (miles)

D – T – G – S – K – D 610

D – T – G – K – S – D 516

D – T – K – S – G – D 588

D – T – K – G – S – D 458

D – T – S – K – G – D 540

D – T – S – G – K – D 504

D – S – T – G – K – D 598

D – S – T – K – G – D 576

D – S – K – T – G – D 682

D – S – G – T – K – D 646

D – G – S – T – K – D 670

D – G – T – S – K – D 728

Euler and Hamilton Paths

An Euler circuit in a graph G is s simple circuit containing every edge of G. An Euler path in G is a simple path containing every edge of G.

Theorem 1

A connected multigraph has an Euler circuit if and only if each of its vertices has even degree.

Theorem 2

A connected multigraph has an Euler path but not Euler circuit if and only if it has exactly two vertices of odd degree.

A path x0, x1, …., xn-1, xn in the graph G = (V, E) is called a Hamilton path if V = {x0, x1, …, xn-1, xn} and xi ≠ xj for 0 ≤ i , j ≤ n. A circuit x0, x1, …, xn-1, x0 (with n > 1) in a graph G = (V, E) is called Hamilton circuit if x0, x1, …, xn-1, xn is a Hamilton path.

Theorem 3

If G is connected simple graph with n vertices where n ≥ 3, then G has a Hamilton circuit if the degree of each vertex is at least n/2.

Exercises:

1. Find the shortest path from a to z and a to x.

a

x

z

b

c

e

d20

42

35

407530

75

45

55

85

2. Solve the traveling salesman problem for the following graph by finding the total weight of all the circuits and determining a circuit with minimum total weight.

a b

c

d

e

3

7 10

1 6

5

94

82