On the determinant of the walk matrix of the rooted product with a path
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2025
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| _version_ | 1866909538456698880 |
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| author | Yan, Zhidan Wang, Wei |
| author_facet | Yan, Zhidan Wang, Wei |
| contents | For an $n$-vertex graph $G$, the walk matrix of $G$, denoted by $W(G)$, is the matrix $[e,A(G)e,\ldots,(A(G))^{n-1}e]$, where $A(G)$ is the adjacency matrix of $G$ and $e$ is the all-ones vector. For two integers $m$ and $\ell$ with $1\le \ell\le (m+1)/2$, let $G\circ P_m^{(\ell)}$ be the rooted product of $G$ and the path $P_m$ taking the $\ell$-th vertex of $P_m$ as the root, i.e., $G\circ P_m^{(\ell)}$ is a graph obtained from $G$ and $n$ copies of the path $P_m$ by identifying the $i$-th vertex of $G$ with the $\ell$-th vertex (the root vertex) of the $i$-th copy of $P_m$ for each $i$. We prove that, $\det W(G\circ P_m^{(\ell)})$ equals $\pm (\det A(G))^{\lfloor\frac{m}{2}\rfloor}(\det W(G))^m$ if $\gcd(\ell,m+1)=1$, and equals 0 otherwise. This extends a recent result established in [Wang et al. Linear Multilinear Algebra 72 (2024): 828--840] which corresponds to the special case $\ell=1$. As a direct application, we prove that if $G$ satisfies $\det A(G)=\pm 1$ and $\det W(G)=\pm 2^{\lfloor n/2\rfloor}$, then for any sequence of integer pairs $(m_i,\ell_i)$ with $\gcd(\ell_i,m_i+1)=1$ for each $i$, all the graphs in the family \begin{equation*}
G\circ P_{m_1}^{(\ell_1)}, (G\circ P_{m_1}^{(\ell_1)})\circ P_{m_2}^{(\ell_2)}, ((G\circ P_{m_1}^{(\ell_1)})\circ P_{m_2}^{(\ell_2)})\circ P_{m_3}^{(\ell_3)},\ldots \end{equation*} are determined by their generalized spectrum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_12130 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the determinant of the walk matrix of the rooted product with a path Yan, Zhidan Wang, Wei Combinatorics 05C50 For an $n$-vertex graph $G$, the walk matrix of $G$, denoted by $W(G)$, is the matrix $[e,A(G)e,\ldots,(A(G))^{n-1}e]$, where $A(G)$ is the adjacency matrix of $G$ and $e$ is the all-ones vector. For two integers $m$ and $\ell$ with $1\le \ell\le (m+1)/2$, let $G\circ P_m^{(\ell)}$ be the rooted product of $G$ and the path $P_m$ taking the $\ell$-th vertex of $P_m$ as the root, i.e., $G\circ P_m^{(\ell)}$ is a graph obtained from $G$ and $n$ copies of the path $P_m$ by identifying the $i$-th vertex of $G$ with the $\ell$-th vertex (the root vertex) of the $i$-th copy of $P_m$ for each $i$. We prove that, $\det W(G\circ P_m^{(\ell)})$ equals $\pm (\det A(G))^{\lfloor\frac{m}{2}\rfloor}(\det W(G))^m$ if $\gcd(\ell,m+1)=1$, and equals 0 otherwise. This extends a recent result established in [Wang et al. Linear Multilinear Algebra 72 (2024): 828--840] which corresponds to the special case $\ell=1$. As a direct application, we prove that if $G$ satisfies $\det A(G)=\pm 1$ and $\det W(G)=\pm 2^{\lfloor n/2\rfloor}$, then for any sequence of integer pairs $(m_i,\ell_i)$ with $\gcd(\ell_i,m_i+1)=1$ for each $i$, all the graphs in the family \begin{equation*} G\circ P_{m_1}^{(\ell_1)}, (G\circ P_{m_1}^{(\ell_1)})\circ P_{m_2}^{(\ell_2)}, ((G\circ P_{m_1}^{(\ell_1)})\circ P_{m_2}^{(\ell_2)})\circ P_{m_3}^{(\ell_3)},\ldots \end{equation*} are determined by their generalized spectrum. |
| title | On the determinant of the walk matrix of the rooted product with a path |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2503.12130 |