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

Abstract

The walk matrix of an n-vertex graph G with adjacency matrix A, denoted by W ( G ) , is [ e , Ae , … , A n − 1 e ] , where e is the all-ones vector. Let G ∘ P m be the rooted product of G and a rooted path P m (taking an endvertex as the root), i.e. G ∘ 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 ∈ { 3 , 4 } , respectively det W ( G ∘ P m ) = ± a 0 ⌊ m 2 ⌋ ( det W ( 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 . In this paper, we verify this conjecture using the technique of Chebyshev polynomials.