ISOMORPHISM OF GRAPH

Two Graphs G₁(V₁,E₁) and G₂(V₂,E₂) are isomorphic if there is a bijective (one to one and onto) mapping f:V₁ –> V₂  that preserves adjacency that is a and b are adjacent in G₁ if and only if f(a) and f(b) are adjacent in G₂, for all a, b belongs to G₁. Such a correspondence f is called graph isomorphism 

Theorem: 
           Two graphs are isomorphic, if and only if their vertices can be labelled in such a way that the corresponding adjacency matrices are equal.

Theorem:    
         Two labeled graphs G and G' with adjacency matrices A and A' respectively are isomorphic, if and if, there exists a permutation matrix   P such that PAP^T =A'

Note:
    A matrix whose rows are the rows of the unit matrix, but not necessarily in their natural order, is called a permutation matrix.