Graph isomorphism testing without numerics for graphs of bounded eigenvalue multiplicity

Abstract

There are several parameterized classes of graphs for which polynomial time isomorphism tests are known. Attempts have been made to develop one conceptionally simple parameterized class of algorithms to solve the graph isomorphism problem for all of these classes. Such unified algorithms have been designed to handle almost all of these classes except for the case of bounded eigenvalue multiplicity. It is shown here that this case can also be handled in a more direct way by discrete methods. The new algorithm uses combinatorics and group theory closely related to the methods used for the other feasible classes of graphs. The classical polynomial time graph isomorphism test of Babai, Grigoriev and Mount for graphs of bounded eigenvalue multiplicity consists of two distinct parts. First, in the linear algebra part, numerical approximations of all eigenvalues and projections of the basis vectors into the eigenspaces are computed. The precision has to be chosen carefully to ensure that it is decidable whether two such projections are equal or have equal length. Also equal angles between such projections have to be recognized. In a second combinatorial and group theoretical part, this information is used to try isomorphisms in the projections and either to combine them to a global isomorphism or to detect that none exists. The numerical part is alien to such a discrete mathematical problem. A direct combinatorial approach is more natural and gives more insight. It is shown that such an approach is indeed pos*Department of Computer Science and Engineering, Pennsylvania State University, University Park, PA 16802, furer@cse .psu. edu and Department of Computer Science, Princeton University, Princeton, NJ 08544. This work is supported in part by NSF grant CCR-9218309 and DIMACS. sible. It is an important step towards one unified algorithm for the graph isomorphism problem for all natural polynomially solvable classes. It helps understanding under which circumstances computationally feasible isomorphism tests are possible.