Haemers' conjecture: an algorithmic perspective

Characterizing graphs by their spectra is a fundamental and challenging problem in spectral graph theory, which has received considerable attention in recent years. A major unsolved conjecture in this area is Haemers' conjecture which states that almost all graphs are determined by their spectra. Despite many efforts, little is known about this conjecture so far. In this paper, we shall consider Haemers' conjecture from an algorithmic perspective. Based on some recent developments in the generalized spectral characterizations of graphs, we propose an algorithm to find all possible generalized cospectral mates for a given $n$-vertex graph $G$, assuming that $G$ is controllable or almost controllable. The experimental results indicate that the proposed algorithm runs surprisingly fast for most graphs with several dozen vertices. Moreover, we observe in the experiment that most graphs are determined by their generalized spectra, e.g., at least 9945 graphs are determined by their generalized spectra among all randomly generated 10,000 graphs on 50 vertices in one experiment. These experimental results give strong evidence for Haemers' conjecture.