The Minimal Total Irregularity of Graphs

TL;DR

This paper investigates the minimal total irregularity in various classes of connected graphs, identifying the graphs with the smallest irregularity measures and proposing open problems for future research.

Contribution

It determines the minimal, second minimal, and third minimal total irregularity values for trees, unicyclic, and bicyclic graphs, advancing understanding of graph irregularity.

Findings

Identified graphs with minimal irregularity in different classes.

Established the order of minimal irregularity values for specific graph types.

Proposed open problems for further exploration of irregularity in graphs.

Abstract

In \cite{2012a}, Abdo and Dimitov defined the total irregularity of a graph $G=(V,E)$ as \hskip3.3cm $\rm irr_{t}$$(G) = \frac{1}{2}\sum_{u,v\in V}|d_{G}(u)-d_{G}(v)|, $ \noindent where $d_{G}(u)$ denotes the vertex degree of a vertex $u\in V$. In this paper, we investigate the minimal total irregularity of the connected graphs, determine the minimal, the second minimal, the third minimal total irregularity of trees, unicyclic graphs, bicyclic graphs on $n$ vertices, and propose an open problem for further research.