### Descriptive complexity of graph spectra

Anuj Dawar, Simone Severini, Octavio Zapata
2019 Annals of Pure and Applied Logic
Two graphs are co-spectral if their respective adjacency matrices have the same multi-set of eigenvalues. A graph is said to be determined by its spectrum if all graphs that are co-spectral with it are isomorphic to it. We consider these properties in relation to logical definability. We show that any pair of graphs that are elementarily equivalent with respect to the three-variable counting first-order logic C 3 are cospectral, and this is not the case with C 2 , nor with any number of
more » ... s if we exclude counting quantifiers. We also show that the class of graphs that are determined by their spectra is definable in partial fixedpoint logic with counting. We relate these properties to other algebraic and combinatorial problems. The spectrum of a graph G is the multi-set of eigenvalues of its adjacency matrix. Even though it is defined in terms of the adjacency matrix of G, the spectrum does not, in fact, depend on the order in which the vertices of G are listed. In other words, isomorphic graphs have the same spectrum. The converse is false: two graphs may have the same spectrum without being isomorphic. Say that two graphs are co-spectral if they have the same spectrum. Our aim in this paper is to study the relationship of this equivalence relation on graphs in relation to a number of other approximations of isomorphism coming from logic, combinatorics and algebra. We also investigate the definability of co-spectrality and related notions in logic. Specifically, we show that for any graph G, we can construct a formula φ G of first-order logic with counting, using only three variables (i.e. the logic C 3 ) so that H |= φ G only if H is co-spectral with G. From this, it follows that elementary equivalence in C 3 refines co-spectrality, a result that also follows from [1] . In contrast, we show that co-spectrality is incomparable with elementary equivalence in C 2 , or with elementary equivalence in L k (first-order logic with k variables but without counting quantifiers) for any k. We show that on strongly regular graphs, co-spectrality exactly co-incides with C 3 -equivalence. ⋆ We thank Aida Abiad, Chris Godsil, Robin Hirsch and David Roberson for fruitful discussions.