The total irregularity of a graph

TL;DR

This paper introduces the total irregularity measure for simple undirected graphs, analyzes its properties, and identifies graphs with maximal irregularity, including the star among trees.

Contribution

It defines a new irregularity measure for graphs and characterizes the graphs that maximize this measure, especially among trees.

Findings

Star graph has maximal irregularity among trees.

Graphs with maximal irregularity are explicitly characterized.

The measure provides a new way to quantify graph irregularity.

Abstract

In this note a new measure of irregularity of a simple undirected graph $G$ is introduced. It is named the total irregularity of a graph and is defined as $\irr_t(G) = 1/2\sum_{u,v \in V(G)} |d_G(u)-d_G(v)|$, where $d_G(u)$ denotes the degree of a vertex $u \in V(G)$. The graphs with maximal total irregularity are determined. It is also shown that among all trees of same order the star graph has the maximal total irregularity.