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Enumeration of cospectral graphs. (English) Zbl 1033.05070
Summary: 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; see C. D. Godsil and B. D. McKay [Aequations Math. 25, 257–268 (1982; Zbl 0527.05051)]. 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 $n<11$ (depending on the matrix). Since the fraction of cospectral graphs on $n$ vertices constructible by GM switching tends to 0 if $n\to \infty$, 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.

##### MSC:
 05C50 Graphs and linear algebra 05C30 Enumeration in graph theory
##### Keywords:
Graph; Eigenvalue; Enumeration