Descriptive complexity of the generalized spectra of graphs

TL;DR

This paper explores the relationship between graph spectra and logical definability, showing that certain logical equivalences imply generalized cospectrality and providing new characterizations of specific graph classes.

Contribution

It establishes a connection between elementary equivalence in $C^3$ logic and generalized cospectrality, and characterizes distance-regular graphs using $C^3$ logic.

Findings

Elementary equivalence in $C^3$ implies generalized cospectrality.

Distance-regular graphs are characterized by $C^3$ logic.

Elementary equivalence in $C^2$ coincides with isomorphism for controllable graphs.

Abstract

Two graphs are cospectral if their respective adjacency matrices have the same multiset of eigenvalues, and generalized cospectral if they are cospectral and so are their complements. We study generalized cospectrality in relation to logical definability. We show that any pair of graphs that are elementary equivalent with respect to the three-variable counting first-order logic $C^3$ are generalized cospectral, and this is not the case with $C^2$, nor with any number of variables if we exclude counting quantifiers. Using this result we provide a new characterization of the well-known class of distance-regular graphs using the logic $C^3$. We also show that, for controllable graphs (it is known that almost all graphs are controllable), the elementary equivalence in $C^2$ coincides with isomorphism.