Isomorphism of graphs-a polynomial test

Introduction-The question whether an isomorphism test for two graphs may be found, which is polynomial in the number of vertices, , n stands open for quite a while now. The purpose of the present article is to answer this question affirmatively by presenting an algorithm which decides whether two graphs are isomorphic or not and showing that the number of n independent elementary operations needed for that decision is bounded from above by ) ) (log ( 2 2 6 n n o . The main ingredient in the construction of the algorithm is a theorem stating that if two real n n× symmetric matrices are such that when raised to any power n k , , 1K = their diagonal elements are identical, the two matrices are identical. This is proven in section 1. In section 2 it is shown how this theorem can be applied to the problem of isomorphism of graphs. This is done essentially by considering the connectivity matrices and asking whether by a properly defined rearrangement of both matrices they can be brought to obey the conditions of theorem I. Section 3 is devoted to the description of an algorithm implementing the ideas presented in section II and to showing that the number of n independent operations needed for the algorithm to answer the question of isomorphism of two graphs is polynomial in n .