Enumeration of cospectral graphs

Authors:
Willem H. Haemers;Edward Spence
Affiliations:
Department of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands;Department of Mathematics, University of Glasgow, Glasgow G12 8QQ, Scotland, UK
Venue:
European Journal of Combinatorics - Special issue on algebraic combinatorics: in memory of J.J. Seidel
Year:
2004

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Abstract

We have enumerated all graphs on at most 11 vertices and determined their spectra with respect to various matrices, such as the adjacency matrix and the Laplacian matrix. We have also counted the numbers for which there is at least one other graph with the same spectrum (a cospectral mate). In addition we consider a construction for pairs of cospectral graphs due to Godsil and McKay, which we call GM switching. It turns out that for the enumerated cases a large part of all cospectral graphs comes from GM switching, and that the fraction of graphs on n vertices with a cospectral mate starts to decrease at some value of n n vertices constructible by GM switching tends to 0 if n → ∞, the present data give some indication that possibly almost no graph has a cospectral mate. We also derive asymptotic lower bounds for the number of graphs with a cospectral mate from GM switching.