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On the singularity and the inverse of 3-colored digraphs

Md Isheteyak Zaffer

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.

On the singularity and the inverse of 3-colored digraphs

Abstract

This article considers the class of connected 3-colored digraphs. Let be a 3-colored digraph and be its adjacency matrix. is said to be non-singular (resp. singular) if is a non-singular (resp. singular) matrix. A connected digraph is k-cyclic if it has vertices and edges. The main objective of this article is to provide a characterization of non-singular 3-colored unicyclic and bicyclic digraphs. If is non-singular and has a diagonal, then 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 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.
Paper Structure (9 sections, 31 theorems, 14 equations, 3 figures)

This paper contains 9 sections, 31 theorems, 14 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Non-singular 3-colored digraphs
  3. Non-singular digraphs in $\mathscr{U}$
  4. Non-singular digraphs in $\mathscr{B}$
  5. $\infty-$Type bicyclic digraph
  6. $\theta-$ Type bicyclic digraph
  7. 3-colored digraphs that has zero diagonal in the inverse of the adjacency matrix
  8. Non-singular digraphs in $\mathscr{B}$ with zero diagonal in the inverse of the adjacency matrix
  9. Inverse of digraphs

Key Result

Lemma 1

(DK1, Lemma 3) Let $G$ be a 3-colored digraph on $n$ vertices. Then where $\mathscr{C}$ is the set of all contributing spanning elementary subgraphs $H$ of $G$, and $|S_H|$ and $|C_H|$ are the number of singular components and cycles in $H$, respectively.

Figures (3)

  • Figure 1: Paths described in Remark \ref{['paths']}.
  • Figure 2: Possible structures corresponding to the cases $i_u',j_v'\in V(P)$ and only $i_u'\in V(P)$.
  • Figure 3:

Theorems & Definitions (73)

  • Lemma 1
  • Remark 2
  • Corollary 3
  • proof
  • Corollary 4
  • proof
  • Lemma 5
  • proof
  • Proposition 6
  • proof
  • ...and 63 more