FRACTIONAL ISOMORPHISM OF GRAPHS

Let the adjacency matrices of graphs G and H be A and B. These graphs are isomorphic provided there is a permutation matrix P with AP=PB, or equivalently, A=PBP(T). If we relax the requirement that P be a permutation matrix, and, instead, require P only to be doubly stochastic, we arrive at two new equivalence relations on graphs: linear fractional isomorphism (when we relax AP=PB) and quadratic fractional isomorphism (when we relax A=PBP(T)). Further, if we allow the two instances of P in A=PBP(T) to be different doubly stochastic matrices, we arrive at the concept of semi-isomorphism. We present necessary and sufficient conditions for graphs to be linearly fractionally isomorphic, we prove that quadratic fractional isomorphism is the same as isomorphism and we relate semi-isomorphism to isomorphism of bipartite graphs.