Counting cospectral graphs obtained via switching
Aida Abiad, Nils Van de Berg, Robin Simoens
TL;DR
The paper develops a general counting framework for graphs that admit a cospectral mate via switching, unifying GM, WQH, AH, and Fano switching schemes. It proves an asymptotically tight formula $\frac{1}{|Aut_Q(\Gamma)|}\,|V_Q|^{n-m}\,g_{n-m}\,(1+o(1))$ for the number of non-isomorphic cospectral mates per switching method, under distinguishing and switching-distinguishing conditions. The results are specialized to several known switching methods (GM$_4$, GM$_6$, GM$_8$, WQH$_6$, WQH$_8$, AH$_6$, Fano) with explicit constants, and complemented by computer enumeration up to $n=10$ that confirms the asymptotics and reveals method-frequency differences. The work advances understanding of the prevalence of cospectral graphs and provides tools toward Haemers' conjecture by offering scalable bounds across switching families. It also discusses limitations, such as applicability to Laplacian or other matrices, and outlines avenues for extending the framework to higher levels and broader switching constructions.
Abstract
Switching is an operation on a graph that does not change the spectrum of the adjacency matrix, thus producing cospectral graphs. An important activity in the field of spectral graph theory is the characterization of graphs by their spectrum. Hence, switching provides a tool for disproving the existence of such a characterization. This paper presents a general framework for counting the number of graphs that have a non-isomorphic cospectral graph through a switching method, expanding on the work by Haemers and Spence [European Journal of Combinatorics, 2004]. Our framework is based on a different counting approach, which allows it to be used for all known switching methods for the adjacency matrix. From this, we derive asymptotic results, which we complement with computer enumeration results for graphs up to $10$ vertices.