On the adjacency matrix of a block graph

Abstract

A block graph is a graph in which every block is a complete graph. Let G be a block graph and let A be the adjacency matrix of G. We first obtain a formula for the determinant of A over reals. It is shown that A is nonsingular over IF2 if and only if the removal of any vertex from G produces a graph with exactly one odd component. A formula for the inverse of A over IF2 is obtained, whenever it exists. We obtain some results for the adjacency matrices over IF2, of claw-free block graphs, which are the same as the line graphs of trees, and for the adjacency matrices of flowers, which are block graphs with only one cut-vertex.