Page 913 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Figure 11.7.11
The following theorem, which uses the matrix S, is helpful for identifying cliques.
THEOREM 11.7.2

Identifying Cliques .
Let be the -th element of . Then a vertex belongs to some clique if and only if

Proof If , then there is at least one 3-step connection from to itself in the modified directed graph determined by S.

Suppose it is . In the modified directed graph, all directed relations are two-way, so we also have the
connections
. But this means that is either a clique or a subset of a clique. In either case, must

belong to some clique. The converse statement, “if belongs to a clique, then ,” follows in a similar manner.

EXAMPLE 5 Using Theorem 11.7.2
Suppose that a directed graph has as its vertex matrix

Then

Because all diagonal entries of are zero, it follows from Theorem 11.7.2 that the directed graph has no cliques.

EXAMPLE 6 Using Theorem 11.7.2
Suppose that a directed graph has as its vertex matrix

Then