Generalized spectral characterization of signed bipartite graphs

TL;DR

This paper establishes conditions under which signed bipartite graphs are uniquely determined by their generalized spectra, extending previous results from signed trees to a broader class of graphs.

Contribution

It generalizes spectral characterization criteria from signed trees to all signed bipartite graphs under specific algebraic conditions.

Findings

Graphs are determined by their spectra if certain algebraic conditions are met.

Extension of spectral characterization from trees to bipartite graphs.

Provides a new criterion involving discriminant and polynomial coefficients.

Abstract

Let $\Sigma$ be an $n$-vertex controllable or almost controllable signed bipartite graph, and let $\Delta_\Sigma$ denote the discriminant of its characteristic polynomial $\chi(\Sigma; x)$. We prove that if (\rmnum{1}) the integer $2^{ -\lfloor n/2 \rfloor }\sqrt{\Delta _{\Sigma}}$ is squarefree, and (\rmnum{2}) the constant term (even $n$) or linear coefficient (odd $n$) of $\chi(\Sigma; x)$ is $\pm 1$, then $\Sigma$ is determined by its generalized spectrum. This result extends a recent theorem of Ji, Wang, and Zhang [Electron. J. Combin. 32 (2025), \#P2.18], which established a similar criterion for signed trees with irreducible characteristic polynomials.