Directed intervals and the dual of a graph

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