An algebraic approach to the graph isomorphism problem

Abstract

This thesis is addressed to the problem of determining graph isomorphism on the computer by exploiting the algebraic properties of the adjacency matrices of the given graphs. More specifically, some results are proven which take advantage of the information provided by the eigenvectors of the adjacency matrices in order to determine graph isomorphism. The proposed methods are capable of either detecting isomorphism or proving non-isomorphism for any given pair of n-point cospectral graphs that are connected and without loops or multiple edges. The most significant contribution of the work is that it provides a means for dealing ,with cospectral graphs that have highly multiple eigenvalues. In Chapter 1 a review of previous isomorphism algorithms is given along with some preliminary notation and definitions. In Chapter 2 an algorithm is proposed which will detect non-isomorphism in order n3logn time at worst for many non-isomorphic cospectral graphs (including some with highly multiple eigenvalues). In Chapter) the algorithm incorporates backtracking to handle isomorphic graphs and the "more difficult" cases of non-isomorphic graphs including strongly regular graphs and graphs of designs. As yet we have not discovered a pair of non-isomorphic cospectral graphs that require more than n "top level" backtracks from this algorithm. For the graphs that require n backtracks, the predicted computation time to prove non-isomorphism is of order n6. Alternatively, the algorithm determines isomorphism of the "non-difficult" class of graphs using 0 backtracks in n5 time at worst. In Chapter 4 a modification of the backtracking algorithm is given fur determining the generators of the automorphism group of a graph. Chapter 5 contains a study of computer timings of the algorithm which confirms that the predicted bounds are in fact too high as n grows. Finally, some directions for future research are suggested in Chapter 6.