A Metzler matrix of a group covering of a digraph

Abstract

We present a decomposition formula for the determinant of a Metzler matrix $A\left(H\right)$ of a group covering H of a digraph D. Furthermore, we introduce an L-function of D with respect to its Metzler matrix $A\left(D\right)$, and present a determinant expression of it. As a corollary, we present a decomposition formula for the determinant of a Metzler matrix $A\left(H\right)$ of a group covering H of D by its L-functions.