Discovering near Symmetry in Graphs

Abstract

Symmetry is a widespread phenomenon that can offer oppor-tunities for powerful exploitation in areas as diverse as molec-ular chemistry, pure mathematics, circuit design, biology and architecture. Graphs are an abstract way to represent rela-tional structures. The search for symmetry in many contexts can thus be reduced to the attempt to find graph automor-phisms. Brendan McKay’s NAUTY system (McKay 1990) is an example of one of the highly successful products of re-search into this task. Erdős and Rényi showed that almost all large graphs are asymmetric, but it is readily observed that many graphs representing structures of real interest contain symmetry. Even more graphs are nearly symmetric, in the sense that to each graph there is a closely similar graph that is symmetric. In this paper we explore the problem of finding near symmetries in graphs and describe the techniques we are developing for performing this task. 1