# Two Irregularity Measures Possessing High Discriminatory Ability

## TL;DR

This paper introduces two new irregularity measures for graphs that uniquely identify antiregular graphs as the most nonregular, enhancing the discriminatory power over existing measures.

## Contribution

The paper proposes two novel irregularity measures that specifically distinguish antiregular graphs as the most nonregular, unlike previous measures.

## Key findings

- Only antiregular graphs are the most nonregular according to the new measures.
- The new measures outperform existing ones in discriminating antiregular graphs.
- The measures have high discriminatory ability for graph irregularity.

## Abstract

An $n$-vertex graph whose degree set consists of exactly $n-1$ elements is called antiregular graph. Such type of graphs are usually considered opposite to the regular graphs. An irregularity measure ($IM$) of a connected graph $G$ is a non-negative graph invariant satisfying the property: $IM(G) = 0$ if and only if $G$ is regular. The total irregularity of a graph $G$, denoted by $irr_t(G)$, is defined as $irr_t(G)= \sum_{\{u,v\} \subseteq V(G)} |d_u - d_v|$ where $V(G)$ is the vertex set of $G$ and $d_u$, $d_v$ denote the degrees of the vertices $u$, $v$, respectively. Antiregular graphs are the most nonregular graphs according to the irregularity measure $irr_t$; however, various non-antiregular graphs are also the most nonregular graphs with respect to this irregularity measure. In this note, two new irregularity measures having high discriminatory ability are devised. Only antiregular graphs are the most nonregular graphs according to the proposed measures.