Descriptive complexity of graph spectra

TL;DR

This paper explores the relationship between graph spectra, logical definability, and graph isomorphism, showing that certain logical equivalences imply co-spectrality and that spectrum-determined graphs are definable in fixed-point logic.

Contribution

It establishes a connection between elementary equivalence in $C^3$ logic and co-spectrality, and shows spectrum-determined graphs are definable in partial fixed-point logic with counting.

Findings

Graphs elementarily equivalent in $C^3$ are co-spectral.

Spectrum-determined graphs are definable in partial fixed-point logic with counting.

The results relate logical definability to algebraic and combinatorial graph properties.

Abstract

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 co-spectral, and this is not the case with $C^2$, nor with any number of variables if we exclude counting quantifiers. We also show that the class of graphs that are determined by their spectra is definable in partial fixed-point logic with counting. We relate these properties to other algebraic and combinatorial problems.