Relationship between Graph Theory and Linear Algebra

By

Shannon Jones

Outline

• Overview of Graph Theory

• Linear Algebra in Graph Theory

• Application of Adjacency Matrices in Graph Theory

• Application of Adjacency Matrices in Network Graph Analysis

Overview

• Graph Theory– Vertices V(G)– Edges E(G)

Linear Algebra in Graph Theory

• Linear Algebra – study of linear sets of equations and their

transformation properties. – Matrices– Isomorphism

Linear Algebra in Graph Theory

Matrices of a Graph– Matrix

– Adjacency Matrix

Linear Algebra in Graph Theory

• Adjacency Matrix- The adjacency matrix for a simple graph G, denoted A(G), is defined as the symmetric matrix whose rows and columns are both indexed by identical ordering of V(G), such that A(G)[u,v] = 1 if u and v are adjacent, otherwise A(G)[u,v]= 0.

• Ex: G= A(G)=

w

v

u

x

u v w xu 0 1 1 0v 1 0 1 0w 1 1 0 1x 0 0 1 0

Linear Algebra in Graph Theory

• Adjacency Matrix- The adjacency matrix of a simple digraph D, denoted A(D), is the matrix whose rows and columns are both indexed by identical orderings of V(G), such that A(D)[u,v]= 1 if there is an edge from u to v, otherwise A(D)[u,v]= 0.

• Ex: G= A(G)=

W

v

u

x

u v w xu 0 0 1 0v 1 0 0 0w 0 1 0 1x 0 0 0 0

Application of Adjacency Matrices in Graph Theory

• Graph Isomorphism– Same adjacency matrix = isomorphic– Different adjacency matrix = may not be isomorphic– Ex:

– Rearrange A(G)-

w y x zw 0 1 1 0y 1 0 0 1x 1 0 0 1z 0 1 1 0

a b c da 0 0 1 1b 0 0 1 1c 1 1 0 0d 1 1 0 0

w z x yw 0 0 1 1z 0 0 1 1x 1 1 0 0y 1 1 0 0

Application of Adjacency Matrices in Graph Theory

• Walks – A sequence of alternating vertices and edges

– Let G be a graph with adjacency matrix A(G). The value of element (A(G))^r [u,v] of the rth power of matrix A(G) equals the number of u-v walks of length r (or directed walks of length r for a digraph).

Application of Adjacency Matrices in Graph Theory

• Walks

• Ex: G=

A(G)= A(G)²= A(G)³=

w

v

u

x

u v w xu 0 1 1 0v 1 0 1 0w 1 1 0 1x 0 0 1 0

u v w xu 2 1 1 1v 1 2 1 1w 1 1 3 0x 1 1 0 1

u v w xu 2 3 4 1v 3 2 4 1w 4 4 2 3x 1 1 3 0

Application of Adjacency Matrices in Network Graph

Analysis

• Social Network Graph– Vertices = people– Edges = relationship between two people

• “married to”, “friends with”, “related to”

– Corresponding adjacency matrix

Application of Adjacency Matrices in Network Graph

Analysis

• Social Network Graph

• Degree Centrality

Bob

Caro l

Ted A lice

Bob Carol Ted AliceBob 0 1 1 0

Carol 1 0 1 0Ted 1 1 0 1

Alice 0 0 1 0

Application of Adjacency Matrices in Network Graph

Analysis

• Social Network Graph

• Directed Graph

Bob

Caro l

Ted A lice

Bob Carol Ted AliceBob 0 1 1 0

Carol 0 0 1 0Ted 1 1 0 0

Alice 0 0 1 0

Application of Adjacency Matrices in Network Graph

Analysis

• Social Network Graph Adjacency Matrix– Matrix Operations

• Transpose- rows and columns exchange = the measure of degrees of the reciprocity of ties within the graph

• Inverse- (original)(inverse)= identity• Addition and Subraction

Application of Adjacency Matrices in Network Graph

Analysis

• Social Network Graph Adjacency Matrix– Key Matrix Operation

• Powers of the Adjacency Matrix– number of walks of different lengths between people– connectivity of a person in the graph

Application of Adjacency Matrices in Network Graph

Analysis

• Social Network Graph Adjacency Matrix– Key Matrix Operation

• Powers of the Adjacency Matrix

1

2

3

4 5

6

7

1

2

3

4

5

6 7

8

9

10

G= H=

1

2

3

4 5

6

7

1 2 3 4 5 6 71 0 1 1 1 1 1 12 1 0 0 0 0 0 03 1 0 0 0 0 0 04 1 0 0 0 0 0 05 1 0 0 0 0 0 06 1 0 0 0 0 0 07 1 0 0 0 0 0 0

A(G)

1 2 3 4 5 6 71 6 0 0 0 0 0 02 0 1 1 1 1 1 13 0 1 1 1 1 1 14 0 1 1 1 1 1 15 0 1 1 1 1 1 16 0 1 1 1 1 1 17 0 1 1 1 1 1 1

A(G)²1 2 3 4 5 6 7

1 0 6 6 6 6 6 62 6 0 0 0 0 0 03 6 0 0 0 0 0 04 6 0 0 0 0 0 05 6 0 0 0 0 0 06 6 0 0 0 0 0 07 6 0 0 0 0 0 0

A(G)³

1

2

3

4

5

6 7

8

9

10

1 2 3 4 5 6 7 8 9 101 0 1 0 0 1 0 0 1 0 02 1 0 1 1 0 0 0 0 0 03 0 1 0 0 0 0 0 0 0 04 0 1 0 0 0 0 0 0 0 05 1 0 0 0 0 1 1 0 0 06 0 0 0 0 1 0 0 0 0 07 0 0 0 0 1 0 0 0 0 08 1 0 0 0 0 0 0 0 1 19 0 0 0 0 0 0 0 1 0 010 0 0 0 0 0 0 0 1 0 0

1 2 3 4 5 6 7 8 9 101 3 0 1 1 0 1 1 0 1 12 0 3 1 0 1 0 0 1 0 03 1 0 1 1 0 0 0 0 0 04 1 0 1 1 0 0 0 0 0 05 0 1 0 0 3 0 0 1 0 06 1 0 0 0 0 1 1 0 0 07 1 0 0 0 0 1 1 0 0 08 0 1 0 0 1 0 0 3 0 09 1 0 0 0 0 0 0 0 1 110 1 0 0 0 0 0 0 0 1 1

1 2 3 4 5 6 7 8 9 101 0 5 0 0 5 0 0 5 0 02 5 1 3 3 0 1 1 0 1 13 0 3 0 0 1 0 0 1 0 04 0 3 0 0 1 0 0 1 0 05 5 0 1 1 0 3 3 0 1 16 0 1 0 0 3 0 0 1 0 07 0 1 0 0 3 0 0 1 0 08 5 0 1 1 0 1 1 0 1 19 0 1 0 0 1 0 0 1 0 010 0 1 0 0 1 0 0 1 0 0

A(H)

A(H)² A(H)³

Application of Adjacency Matrices in Network Graph

Analysis

• Significance– Marketers– Social Network Websites

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