Proof of a conjecture on the determinant of walk matrix of rooted product with a path
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866916420970872832 |
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| author | Wang, Wei Yan, Zhidan Mao, Lihuan |
| author_facet | Wang, Wei Yan, Zhidan Mao, Lihuan |
| contents | The walk matrix of an $n$-vertex graph $G$ with adjacency matrix $A$, denoted by $W(G)$, is $[e,Ae,\ldots,A^{n-1}e]$, where $e$ is the all-ones vector. Let $G\circ P_m$ be the rooted product of $G$ and a rooted path $P_m$ (taking an endvertex as the root), i.e., $G\circ P_m$ is a graph obtained from $G$ and $n$ copies of $P_m$ by identifying each vertex of $G$ with an endvertex of a copy of $P_m$. Mao-Liu-Wang (2015) and Mao-Wang (2022) proved that, for $m=2$ and $m\in\{3,4\}$, respectively$$\det W(G\circ P_m)=\pm a_0^{\lfloor\frac{m}{2}\rfloor}(\det W(G))^m,$$ where $a_0$ is the constant term of the characteristic polynomial of $G$. Furthermore, Mao-Wang (2022) conjectured that the formula holds for any $m\ge 2$. In this note, we verify this conjecture using the technique of Chebyshev polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_07229 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Proof of a conjecture on the determinant of walk matrix of rooted product with a path Wang, Wei Yan, Zhidan Mao, Lihuan Combinatorics 05C50 The walk matrix of an $n$-vertex graph $G$ with adjacency matrix $A$, denoted by $W(G)$, is $[e,Ae,\ldots,A^{n-1}e]$, where $e$ is the all-ones vector. Let $G\circ P_m$ be the rooted product of $G$ and a rooted path $P_m$ (taking an endvertex as the root), i.e., $G\circ P_m$ is a graph obtained from $G$ and $n$ copies of $P_m$ by identifying each vertex of $G$ with an endvertex of a copy of $P_m$. Mao-Liu-Wang (2015) and Mao-Wang (2022) proved that, for $m=2$ and $m\in\{3,4\}$, respectively$$\det W(G\circ P_m)=\pm a_0^{\lfloor\frac{m}{2}\rfloor}(\det W(G))^m,$$ where $a_0$ is the constant term of the characteristic polynomial of $G$. Furthermore, Mao-Wang (2022) conjectured that the formula holds for any $m\ge 2$. In this note, we verify this conjecture using the technique of Chebyshev polynomials. |
| title | Proof of a conjecture on the determinant of walk matrix of rooted product with a path |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2208.07229 |