Primary decomposition theorem and generalized spectral characterization of graphs
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912333047005184 |
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| author | Guo, Songlin Wang, Wei Wang, Wei |
| author_facet | Guo, Songlin Wang, Wei Wang, Wei |
| contents | Suppose $G$ is a controllable graph of order $n$ with adjacency matrix $A$. Let $W=[e,Ae,\ldots,A^{n-1}e]$ ($e$ is the all-one vector) and $Δ=\prod_{i>j}(α_i-α_j)^2$ ($α_i$'s are eigenvalues of $A$) be the walk matrix and the discriminant of $G$, respectively. Wang and Yu \cite{wangyu2016} showed that if
$$θ(G):=\gcd\{2^{-\lfloor\frac{n}{2}\rfloor}\det W,Δ\} $$
is odd and squarefree, then $G$ is determined by its generalized spectrum (DGS). Using the primary decomposition theorem, we obtain a new criterion for a graph $G$ to be DGS without the squarefreeness assumption on $θ(G)$. Examples are further given to illustrate the effectiveness of the proposed criterion, compared with the two existing methods to deal with the difficulty of non-squarefreeness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12932 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Primary decomposition theorem and generalized spectral characterization of graphs Guo, Songlin Wang, Wei Wang, Wei Combinatorics Suppose $G$ is a controllable graph of order $n$ with adjacency matrix $A$. Let $W=[e,Ae,\ldots,A^{n-1}e]$ ($e$ is the all-one vector) and $Δ=\prod_{i>j}(α_i-α_j)^2$ ($α_i$'s are eigenvalues of $A$) be the walk matrix and the discriminant of $G$, respectively. Wang and Yu \cite{wangyu2016} showed that if $$θ(G):=\gcd\{2^{-\lfloor\frac{n}{2}\rfloor}\det W,Δ\} $$ is odd and squarefree, then $G$ is determined by its generalized spectrum (DGS). Using the primary decomposition theorem, we obtain a new criterion for a graph $G$ to be DGS without the squarefreeness assumption on $θ(G)$. Examples are further given to illustrate the effectiveness of the proposed criterion, compared with the two existing methods to deal with the difficulty of non-squarefreeness. |
| title | Primary decomposition theorem and generalized spectral characterization of graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.12932 |