Rank of weighted digraphs with blocks

TL;DR

This paper investigates the rank of weighted digraphs by analyzing their blocks, introducing classes of digraphs with ranks determined by subdigraphs, and applying these concepts to specific graph types.

Contribution

It introduces new classes of digraphs, such as r2- and r0-digraphs, for which the rank can be explicitly calculated from block subdigraphs.

Findings

Exact rank formulas for r2- and r0-digraphs

Rank determination for directed trees and simple block graphs

Extension of rank analysis to specific classes of digraphs

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.