Fractional isomorphism of graphs

Abstract 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.