A Note on the Determinant of a Special Class of QQ-Walk Matrices

Abstract

For a graph G of order n, its Q-walk matrix is defined by $W_{Q}(G)=\left[e, Q e, \cdots, Q^{n-1} e\right]$, where Q is the signless Laplacian matrix of G and e denotes the all-one column vector. Let $G \circ P_{k}$ represent the rooted product graph of G and a path P_k. In this note, we establish the relationship between determinants of $W_{Q}(G)$ and $ W_{Q}\left(G \circ P_{k}\right)$ for k=2,3.

Keywords

Signless Laplacian matrix / Q-walk matrix / rooted product graph / determinant