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) &isin R .
Multiplication of boolean matrices is done in the same way as ordinary matrix multiplication, but using &and for · and &or for + .
| Property: | Matrix: |
| Identity, R0 | In (identity matrix) |
| Inverse, R-1 or &Gamma-1 | MT |
| Reflexive | I &sube M |
| Symmetric | M = MT |
| Transitive | M2 &sube M |
| Antisymmetric | M &cap MT &sube In |
| Paths of length n | Mn |
| Transitive closure &Gamma+ | &cupi=1n Mi |
| Reflexive transitive closure &Gamma* | &cupi=0n Mi |
Example: Let the set S be basic blocks of a program and &Gamma be transfers of control between blocks.