Generalized spectral characterization of signed bipartite graphs

Abstract

Let $\hat{I}\pounds$ be an $n$-vertex controllable or almost controllable signed bipartite graph, and let $\hat{I}{”}_{\hat{I}}\pounds$ denote the discriminant of its characteristic polynomial $\ddot{I}‡(\hat{I}\pounds ;x)$. We prove that if (\rmnum{1}) the integer $2^{-\lfloor n/2\rfloor }\sqrt{\hat{I}{”}_{\hat{I}}\pounds }$ is squarefree, and (\rmnum{2}) the constant term (even $n$) or linear coefficient (odd $n$) of $\ddot{I}‡(\hat{I}\pounds ;x)$ is $\pm 1$, then $\hat{I}\pounds$ 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.