Logical aspects of isomorphism of controllable graphs and cospectrality of distance-regularized graphs

TL;DR

This paper explores the logical definability of isomorphism and cospectrality in controllable and distance-regularized graphs, unifying algebraic, spectral, and logical approaches.

Contribution

It introduces a logical framework to analyze graph equivalences, extending existing algebraic and spectral characterizations for specific graph classes.

Findings

Logical definability characterizes graph isomorphism and cospectrality.

Unified approach bridges algebraic, spectral, and logical methods.

Extends known results to broader classes of graphs.

Abstract

We consider isomorphism of controllable graphs and cospectrality of distance-regularized graphs (which are known to be distance-regular or distance-biregular) in relation to logical definability. While most characterizations of these equivalence relations for such graph classes are of algebraic and spectral flavor, here we inject tools from first-order logic, extending and unifying several existing results.