Directed intervals and the dual of a graph

We present a class of graphs whose adjacency matrices are nonsingular with integral inverses, denoted h-graphs. For the h-graph G with adjacency matrix M, we consider the problem of identifying exactly when M-1 is signature-similar to the adjacency matrix of another h-graph, G+. When this holds, G+ is a type of graph-inverse of G, which is known as the dual of G. We present necessary and sufficient conditions for the existence of G+. As an application, we provide a characterization of all dual pairs G and G+ where both graphs are unicyclic. This characterization allows us to identify those h-graphs G that are unicyclic and self-dual.