Factorization of invariant polynomials and generalized spectral characterizations of graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913093244682240 |
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| author | Wang, Wei Tang, Quanyu Wang, Wei |
| author_facet | Wang, Wei Tang, Quanyu Wang, Wei |
| contents | 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). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_02711 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Factorization of invariant polynomials and generalized spectral characterizations of graphs Wang, Wei Tang, Quanyu Wang, Wei Combinatorics 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). |
| title | Factorization of invariant polynomials and generalized spectral characterizations of graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.02711 |