Graph isomorphism and multivariate graph spectrum

TL;DR

This paper introduces a new criterion based on the generalized block Laplacian spectrum to distinguish almost all graphs, addressing limitations of existing algorithms and supporting Haemers' conjecture about spectral identification.

Contribution

It proposes a novel spectral criterion that can distinguish graphs beyond the capabilities of the 2-dimensional Weisfeiler-Lehman algorithm, advancing graph isomorphism research.

Findings

Almost all graphs are distinguishable by the proposed spectral criterion.

Supports Haemers' conjecture that almost all graphs are identified by their spectrum.

Provides a new approach to graph isomorphism using generalized block Laplacian spectrum.

Abstract

We provide a criterion to distinguish two graphs which are indistinguishable by $2$-dimensional Weisfeiler-Lehman algorithm for almost all graphs. Haemers conjectured that almost all graphs are identified by their spectrum. Our approach suggests that almost all graphs are identified by their generalized block Laplacian spectrum.