Bounds and Optimal Results for the Total Irregularity Measure

TL;DR

The paper provides a comprehensive survey of the total irregularity measure $irr_t$, highlighting its foundational relation $irr_t(G)=irr(G)+irr(\overline{G})$ and its link to the Gini coefficient via $irr_t(G)=2nm\cdot\zeta(G)$, with particular emphasis on molecular graphs. It catalogs extremal results across diverse graph families—trees, unicyclic, bicyclic, antiregular graphs, polyomino chains, and fixed-order graphs with cyclomatic number—deriving exact maximum and minimum values and identifying the corresponding extremal structures. A broad set of lower and upper bounds is organized around standard graph invariants such as $irr$, degree sequences, Zagreb indices, diameter, and the Harary-Albertson index, along with how these bounds behave under common graph operations. The article concludes with open problems and conjectures that chart clear directions for future work in extremal irregularity measures within molecular graph theory and related combinatorics.

Abstract

A (molecular) graph in which all vertices have the same degree is known as a regular graph. According to Gutman, Hansen, and Mélot [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.

Figures (2)

• Figure 1: The $n$-order antiregular graphs $A_n$ (including the dotted edges) and $\overline{A}_n$ (excluding the dotted edges) are shown for even and odd orders, depicted on the left and right, respectively.
• Figure 2: Two graphs referred in Theorem \ref{['Nasiri-IJMC-18-min-thm2']}.

Theorems & Definitions (39)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10