A note on path-zero graphs (Q1199634)

Let \(A_ k\) denote the (1,0)-adjacency matrix of the path of length \(k\). Let \(P_ k(\lambda):=\text{det}(\lambda I_ k-A_ k)\) denote the characteristic polynomial of \(A_ k\). The \(n\times n\) matrix \(P_ k(X_ n)\) is obtained from \(P_ k(\lambda)\) by substituting for \(\lambda\) the (1,0)-adjacency matrix \(X_ n\) of any connected graph on \(n\) vertices. The matrix equation \(P_ k(X_ n)=O_ n\) is solved by the adjacency matrix of any Coxeter-Dynkin graph on \(n\) vertices, for \(k+1\) any \(\text{Z}^ +\)-multiple of the Coxeter number of that graph. This result was obtained in \textit{R. B. Bapat} and \textit{A. K. Lal} [Linear Algebra Appl. 149, 125-149 (1991; Zbl 0764.05056)]. The proof presented here uses methods of Coxeter groups.