Rank of weighted digraphs with blocks

Abstract

Let $G$ be a digraph and $r(G)$ be its rank. Many interesting results on the rank of an undirected graph appear in the literature, but not much information about the rank of a digraph is available. In this article, we study the rank of a digraph using the ranks of its blocks. In particular, we define classes of digraphs, namely $r_2$-digraph, and $r_0$-digraph, for which the rank can be exactly determined in terms of the ranks of subdigraphs of the blocks. Furthermore, the rank of directed trees, simple biblock graphs, and some simple block graphs are studied.