Proof of a conjecture on the determinant of the walk matrix of rooted product with a path

The walk matrix of an n-vertex graph G with adjacency matrix A, denoted by W(G) W(G), is [e, A(e), ..., A(n-1)e] [e, A(e), ... , A(n-1)e], where e is the all-ones vector. Let G (sic) P-m be the rooted product of G and a rooted path P-m (taking an endvertex as the root), i.e. G (sic) P-m is a graph obtained from G and n copies of P-m by identifying each vertex of G with an endvertex of a copy of P-m. Mao et al. [A new method for constructing graphs determined by their generalized spectrum. Linear Algebra Appl. 2015;477:112-127.] and Mao and Wang [Generalized spectral characterization of rooted product graphs. Linear Multilinear Algebra. 2022. DOI:10.1080/03081087.2022.2098226.] proved that, for m = 2 and m is an element of {3,4, respectively det W(G (sic) P-m) = +/- a(0)([m/2]) (detW(G))m, where a(0) is the constant term of the characteristic polynomial of G. Furthermore, in the same paper, Mao and Wang conjectured that the formula holds for any m >= 2 m >= 2. In this paper, we verify this conjecture using the technique of Chebyshev polynomials.