Isocodal and isospectral points, edges, and pairs in graphs and how to cope with them in computerized symmetry recognition

It is demonstrated that in certain graphs isospectral edges and pairs exist, in analogy to the well-known isospectral points. A pair is any relationship between two vertices (an edge is thus a special kind of a pair), and isospectral pairs are pairs which, when arbitrarily but identically perturbed, always yield isospectral graphs. The significance of isospectral points, edges, and pairs is that computer programs for symmetry perception and for graph isomorphism testing tend to encounter difficulties when processing graphs containing such features; they tend to take isospectrality for equivalence by symmetry. It is shown how in the authors’ programs TOPSYM and MATSYM these difficulties are overcome by using the newly developed “class matrix procedure”. © 1991, American Chemical Society. All rights reserved.