Bounds and Optimal Results for the Total Irregularity Measure

TL;DR

This paper reviews existing bounds and optimal results for the total irregularity measure in graphs, highlighting its relevance in molecular graph analysis and discussing open problems in the field.

Contribution

It compiles and analyzes known bounds and optimal results for the total irregularity measure, and presents open problems for future research.

Findings

Summarizes existing bounds for the total irregularity measure.

Identifies open problems in the optimization of the measure.

Highlights relevance to molecular graph analysis.

Abstract

A (molecular) graph in which all vertices have the same degree is known as a regular graph. According to Gutman, Hansen, and M\'elot [J. Chem. Inf. Model. 45 (2005) 222-230], it is of interest to measure the irregularity of nonregular molecular graphs both for descriptive purposes and for QSAR/QSPR studies. The graph invariants that can be used to measure the irregularity of graphs are referred to as irregularity measures. One of the well-studied irregularity measures is the ``total irregularity'' measure, which was introduced about a decade ago. Bounds and optimization problems for this measure have already been extensively studied. A considerable number of existing results (concerning this measure) also hold for molecular graphs; particularly, the ones regarding lower bounds and minimum values of the mentioned measure. The primary objective of the present review article is to collect the existing bounds and optimal results concerning the total irregularity measure. Several open problems related to the existing results on the total irregularity measure are also given.