Graphs, relations and matrices

- Section 7.4

Overview

- Graph structures are valuable because they can

represent relationships among pairs of objects,

and they remain simple in structure even when the

number of objects is large. In application

problems, it becomes important to use a computer

to analyze graph properties, so we need a

representation of a graph that a computer can

understand.

One representation for a graph

Example

Practice Problem 1

- Write the adjacency matrices of each of the

following graphs. Make clear which vertex

corresponds to which row/column. In each graph,

what is the value of M12? Of M34? Of M21?

Directed Graphs

- Note that all of our examples involve symmetric

matrices because the number in Mab and Mba must

be the same for any graph. However, in some

applications, this is not the case.

Directed Graphs

- A directed graph, like a graph, consists of a set

V of vertices and a set E of edges. Each edge is

associated with an ordered pair of vertices

called its endpoints. In other words, a directed

graph is the same as a graph, but the edges are

described as ordered pairs rather than unordered

pairs. - If the endpoints for edge e are a and b in that

order, we say e is an edge from a to b, and in

the diagram we draw the edge as a straight or

curved arrow from a to b. - For a directed graph, we use (a, b) rather than

a, b to indicate an edge from a to b. This

emphasizes that the edge is an ordered pair, by

using the usual notation for ordered pairs.

More definitions and terms

- A walk in a directed graph is a sequence v1e1v2e2

. . . vnenvn1 of alternating vertices and edges

that begins and ends with a vertex, and where

each edge in the list lies between its endpoints

in the proper order. If there is no chance of

confusion, we omit the edges when we describe a

walk. - The adjacency matrix for a directed graph with

vertices v1, v2, . . . , vn is the n n matrix

where Mij (the entry in row i , column j) is the

number of edges from vertex vi to vertex vj.

Example

- Consider a two-player game where there is a

single pile of 10 stones and each player may

remove one or two stones at a time on his or her

turn. If we use a node for each state of the

game and an edge to denote a move, we get the

directed graph below

Questions

- Why is it important to use a directed graph in

modeling this game? - Find the adjacency matrix for this directed graph.

Matrix arithmetic

- Matrices can be multiplied and added using some

standard mathematical rules. The surprising thing

is that these standard operations, when applied

to the adjacency matrix of a graph, have a real

interpretation in terms of the graph properties

we already know.

What does M2 represent?

- Given the adjacency matrix M for a directed graph

G, the arithmetic operation M M has an

interpretation in G. Lets see if we can figure

out what it is in this example.

Matrix multiplication

- To compute the (2,3)-entry in M2, we multiply Row

2 of M times Column 3 of M as follows

Matrix multiplication

- If we complete the multiplication (every row

times every column), we get the result at the

right. What do these numbers mean in terms of

the original graph?

Interpretation of Mk

- Theorem. Let M be the adjacency matrix of a

directed graph G with vertex set 1, 2, 3, . . .

, n. The row i, column j entry of Mk counts the

number of k-step walks from node i to node j in

the graph G.

Interpretation of the sum M M2 Mk

- We find the sum of two matrices by adding entries

in the same position. This gives us the

following extension of our theorem. - Corollary. Let M be the adjacency matrix of a

directed graph G with vertex set 1, 2, 3, . . .

, n. The row i, column j entry of M M2

Mk counts the number of walks from node i to node

j in the graph G of length k or less.

Example

Connection to Relations

- Note that a directed graph looks exactly the same

as a one-set arrow diagram for a relation R on a

set A. This is no coincidence!!

Binary Relations, Directed Graphs, and Adjacency

Matrices

- For a relation R on the set A 1, 2, 3, . . .

, n, the following statements are equivalent for

all a, b ? A - (a, b) ? R (which we write sometimes as aRb).
- There is a directed edge from node a to node b in

the graph of R. - There is a 1 in the row a, column b entry of the

adjacency matrix for R.

Examples

Solutions

Boolean Operations and Composition of Relations

- We can find a connection between matrix

arithmetic and composition of relations as long

as we use Boolean arithmetic in our

computations

Boolean matrix multiplication

- In this example, we multiply A A in the same

way as before, except we use the Boolean addition

and multiplication among the entries. To

distinguish this from A2, we denote this A(2).

Try it before checking the next slide for the

answer!

Boolean matrix multiplication

- Take a moment to find the composition R ? R, and

write the adjacency matrix for this new relation.

Compare to A(2)!

Composition and Boolean matrix multiplication

- Theorem. If R is a binary relation on a set A

with adjacency matrix M, then the matrix M(2) is

the adjacency matrix for the relation R ? R on

the set A.