On the Determinant of the Q-walk Matrix of Rooted Product with a Path

Let G be an n-vertex graph and Q(G) be its signless Laplacian matrix. The Q-walk matrix of G, denoted by \(W_Q(G)\) $W_Q(G)$ , is \([e,Q(G)e,\ldots ,Q^{n-1}(G)e]\) $[e,Q(G)e,\dots ,Q^{n-1}(G)e]$ , where e is the all-one vector. Let \(G\circ P_m\) $G\circ P_m$ be the graph obtained from G and n copies of the path \(P_m\) $P_m$ by identifying the i-th vertex of G with an endvertex of the i-th copy of \(P_m\) $P_m$ for each i. We prove that, \(\begin{aligned} \det W_Q(G\circ P_m)=\pm (\det Q(G))^{m-1}(\det W_Q(G))^m \end{aligned}\) $$\begin{array}{c}\hfill det{W}_Q(G\circ P_m)=\pm {(detQ(G))}^{m-1}{(det{W}_Q(G))}^m\end{array}$$ holds for any \(m\ge 2\) $m\ge 2$ . This gives a signless Laplacian counterpart of the following recently established identity (Wang et al. in Linear Multilinear Algebra 72:828–840, 2024. https://doi.org/10.1080/03081087.2023.2165612): \(\begin{aligned} \det W_A(G\circ P_m)=\pm (\det A(G))^{\lfloor \frac{m}{2}\rfloor }(\det W_A(G))^m, \end{aligned}\) $$\begin{array}{c}\hfill det{W}_A(G\circ P_m)=\pm {(detA(G))}^{\lfloor \frac{m}{2}\rfloor }{(det{W}_A(G))}^m,\end{array}$$ where A(G) is the adjacency matrix of G and \(W_A(G)=[e,A(G)e,\ldots ,A^{n-1}(G)e]\) $W_A(G)=[e,A(G)e,\dots ,A^{n-1}(G)e]$ . We also propose a conjecture to unify the above two equalities.