A general formula for walk determinants of rooted products with applications to DGS-graph constructions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866912801724825600 |
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| author | Wang, Wei Shen, Jie Mao, Lihuan |
| author_facet | Wang, Wei Shen, Jie Mao, Lihuan |
| contents | For an $n$-vertex graph $G$, and a rooted graph $H^{(v)}$ with $v$ as the root, the rooted product graph $G\circ H^{(v)}$ is obtained from $G$ and $n$ copies of $H$ by identifying the root of the $i$th copy of $H$ with the $i$th vertex of $G$ for each $i$. As a refinement of the controllability criterion of $G\circ H^{(v)}$ obtained recently by Shan and Liu (2025), we obtain an explicit formula for the determinant of the walk matrix of $G\circ H^{(v)}$. Furthermore, for an important family of graphs $\mathcal{F}$ that are determined by their generalized spectrum (DGS), we introduce the concept of $\mathcal{F}$-preservers and provide a sufficient condition for a rooted graph to be an $\mathcal{F}$-preserver. A list of $\mathcal{F}$-preservers of small order is provided, which leads to many new infinite families of DGS-graphs using rooted products. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01542 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A general formula for walk determinants of rooted products with applications to DGS-graph constructions Wang, Wei Shen, Jie Mao, Lihuan Combinatorics 05C50 For an $n$-vertex graph $G$, and a rooted graph $H^{(v)}$ with $v$ as the root, the rooted product graph $G\circ H^{(v)}$ is obtained from $G$ and $n$ copies of $H$ by identifying the root of the $i$th copy of $H$ with the $i$th vertex of $G$ for each $i$. As a refinement of the controllability criterion of $G\circ H^{(v)}$ obtained recently by Shan and Liu (2025), we obtain an explicit formula for the determinant of the walk matrix of $G\circ H^{(v)}$. Furthermore, for an important family of graphs $\mathcal{F}$ that are determined by their generalized spectrum (DGS), we introduce the concept of $\mathcal{F}$-preservers and provide a sufficient condition for a rooted graph to be an $\mathcal{F}$-preserver. A list of $\mathcal{F}$-preservers of small order is provided, which leads to many new infinite families of DGS-graphs using rooted products. |
| title | A general formula for walk determinants of rooted products with applications to DGS-graph constructions |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2601.01542 |