Factorization of invariant polynomials and generalized spectral characterizations of graphs

Abstract

The problem of characterizing graphs by their generalized spectra has received significant attention in recent years. This paper provides a complete proof of a conjecture proposed by Wang, Wang, and Zhu (European J. Combin., 2023), which asserts that the square-root polynomial of the invariant polynomial $Φ_p(G;x) \in \mathbb{F}_p[x]$ can replace its square-free part to yield a more effective criterion for a graph to be determined by its generalized spectrum (DGS). A key ingredient of our proof is a novel algebraic factorization: we show that the polynomial $Φ_p(G;x)$ is the product of the characteristic polynomials of the adjacency operator restricted to the left null space of the walk matrix and its radical, respectively. Based on this refined DGS-criterion, a broad family of DGS-graphs is constructed via rooted products, significantly generalizing the recent result of Wang, Shen, and Mao (Discrete Appl. Math., 2026).