Certain Types of Total Irregularities of Graphs and Digraphs

TL;DR

This paper introduces new concepts related to total irregularity in graphs and digraphs, including edge-transformation, edge-joint, and irregularity measures based on in-degree and out-degree, expanding the understanding of graph irregularities.

Contribution

It presents novel invariants and concepts such as edge-transformation, edge-joint, and irregularity measures for undirected and directed graphs, initiating their study.

Findings

Defined total irregularity for undirected graphs and digraphs.

Introduced edge-transformation and edge-joint concepts.

Initiated analysis of these irregularity parameters.

Abstract

The total irregularity of a simple undirected graph $G$ is denoted by $irr_t(G)$ and is defined as $irr_t(G) = \frac{1}{2}\sum\limits_{u,v \in V(G)}|d(u) - d(v)|$. In this paper, the concept called edge-transformation in relation to total irregularity of simple undirected graphs with at least one cut edge is introduced. We also introduce the concept of an edge-joint between two simple undirected graphs. We also introduce the concept of total irregularity in respect of in-degree and out-degree in simple directed graphs. These invariants are called total in-irregularity and total out-irregularity respectively. In this paper, we initiate a study on these parameters of given simple undirected graphs and simple digraphs.