On the determinant of the $Q$-walk matrix of rooted product with a path
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| Format: | Preprint |
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2023
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| _version_ | 1866912055967088640 |
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| author | Yan, Zhidan Mao, Lihuan Wang, Wei |
| author_facet | Yan, Zhidan Mao, Lihuan Wang, Wei |
| contents | Let $G$ be an $n$-vertex graph and $Q(G)$ be its signless Laplacian matrix. The $Q$-walk matrix of $G$, denoted by $W_Q(G)$, is $[e,Q(G)e,\ldots,Q^{n-1}(G)e]$, where $e$ is the all-one vector. Let $G\circ P_m$ be the graph obtained from $G$ and $n$ copies of the path $P_m$ by identifying the $i$-th vertex of $G$ with an endvertex of the $i$-th copy of $P_m$ for each $i$. We prove that, $$\det W_Q(G\circ P_m)=\pm (\det Q(G))^{m-1}(\det W_Q(G))^m$$ holds for any $m\ge 2$. This gives a signless Laplacian counterpart of the following recently established identity [17]: $$\det W_A(G\circ P_m)=\pm (\det A(G))^{\lfloor\frac{m}{2}\rfloor}(\det W_A(G))^m,$$ where $A(G)$ is the adjacency matrix of $G$ and $W_A(G)=[e,A(G)e,\ldots,A^{n-1}(G)e]$. We also propose a conjecture to unify the above two equalities. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_01123 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the determinant of the $Q$-walk matrix of rooted product with a path Yan, Zhidan Mao, Lihuan Wang, Wei Combinatorics 05C50 Let $G$ be an $n$-vertex graph and $Q(G)$ be its signless Laplacian matrix. The $Q$-walk matrix of $G$, denoted by $W_Q(G)$, is $[e,Q(G)e,\ldots,Q^{n-1}(G)e]$, where $e$ is the all-one vector. Let $G\circ P_m$ be the graph obtained from $G$ and $n$ copies of the path $P_m$ by identifying the $i$-th vertex of $G$ with an endvertex of the $i$-th copy of $P_m$ for each $i$. We prove that, $$\det W_Q(G\circ P_m)=\pm (\det Q(G))^{m-1}(\det W_Q(G))^m$$ holds for any $m\ge 2$. This gives a signless Laplacian counterpart of the following recently established identity [17]: $$\det W_A(G\circ P_m)=\pm (\det A(G))^{\lfloor\frac{m}{2}\rfloor}(\det W_A(G))^m,$$ where $A(G)$ is the adjacency matrix of $G$ and $W_A(G)=[e,A(G)e,\ldots,A^{n-1}(G)e]$. We also propose a conjecture to unify the above two equalities. |
| title | On the determinant of the $Q$-walk matrix of rooted product with a path |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2309.01123 |