On the singularity and the inverse of 3-colored digraphs

TL;DR

This paper characterizes the invertibility and inverse properties of connected 3-colored digraphs, focusing on unicyclic and bicyclic cases, and explores when their inverse matrices correspond to similarly colored digraphs.

Contribution

It provides a new characterization of non-singular 3-colored unicyclic and bicyclic digraphs and analyzes the structure of their inverse matrices.

Findings

Identified all non-singular 3-colored unicyclic and bicyclic digraphs.

Determined conditions under which the inverse matrix has zero diagonal.

Classified bicyclic and unicyclic 3-colored digraphs with inverses that are also 3-colored digraphs.

Abstract

This article considers the class of connected 3-colored digraphs. Let $G$ be a 3-colored digraph and $A(G)$ be its adjacency matrix. $G$ is said to be non-singular (resp. singular) if $A(G)$ is a non-singular (resp. singular) matrix. A connected digraph is k-cyclic if it has $n$ vertices and $n+k-1$ edges. The main objective of this article is to provide a characterization of non-singular 3-colored unicyclic and bicyclic digraphs. If $A(G)$ is non-singular and $A(G)^{-1}$ has a $zero$ diagonal, then $A(G)^{-1}$ can be realized as the adjacency matrix of a digraph with complex weights. Therefore, we also identify all 3-colored bicyclic digraphs such that the diagonal of $A(G)^{-1}$ is zero. Furthermore, we study the invertibility of these digraphs and identify all those bicyclic 3-colored digraphs whose inverse is also a 3-colored digraph. We conduct the same study for the class of unicyclic 3-colored digraphs.