  A block graph is a graph in which every block is a complete graph. Let $G$ be
a block graph and let $A(G)$ be its (0,1)-adjacency matrix. Graph $G$ is called
nonsingular (singular) if $A(G)$ is nonsingular (singular). An interesting open
problem, proposed in 2013 by Bapat and Roy, is to characterize nonsingular
block graphs. In this article, we present some classes of nonsingular and
singular block graphs and related conjectures.
