Determinantal properties of Boolean graphs using recursive approach

Abstract

The aim of this paper is to study the determinant and inverse of the adjacency matrices of weighted and directed versions of Boolean graphs. Our approach is recursive. We describe the adjacency matrix of a weighted Boolean graph in terms of the adjacency matrix of a smaller-sized weighted Boolean graph. This allows us to compute the determinant and inverse of the adjacency matrix of a weighted Boolean graph recursively. In particular, we show that the determinant of a directed Boolean graph is 1. Further, using a classical theorem of Cayley which expresses the determinant of any skew-symmetric matrix as a square of its Pfaffian, we show that for any directed Boolean graph, the characteristic polynomial has all its even degree coefficients strictly positive with the odd ones being zero.