cs375 p. 278

Boolean Matrix Representation of Graph

A relation R or graph on a finite set can be expressed as a boolean matrix M where:

M[i, j] = 1 iff (i, j) ∈ R .

Multiplication of boolean matrices is done in the same way as ordinary matrix multiplication, but using ∧ for · and ∨ for + .

Property: Matrix:

Identity, R⁰ I_n (identity matrix)

Inverse, R^-1 or Γ^-1 M^T

Reflexive I ⊆ M

Symmetric M = M^T

Transitive M² ⊆ M

Antisymmetric M ∩ M^T ⊆ I_n

Paths of length n Mⁿ

Transitive closure Γ⁺ ∪_i=1ⁿ Mⁱ

Reflexive transitive closure Γ^* ∪_i=0ⁿ Mⁱ

Example: Let the set S be basic blocks of a program and Γ be transfers of control between blocks.