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 WQ(G), is [e,Q(G)e, . . . ,Q(n-1)(G)e], where e is the all-one vector. Let G circle 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, detW(Q)(G circle P-m)=+/-(detQ(G))(m-1)(detW(Q)(G))(m) holds for any m >= 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): detW(A)(G circle P-m)=+/-(detA(G))((sic)m/2(sic))(detWA(G))(m), where A(G) is the adjacency matrix of G and W-A(G)=[e,A(G)e, . . . ,A(n-1)(G)e]. We also propose a conjecture to unify the above two equalities.