The Total Irregularity of Graphs under Graph Operations

TL;DR

This paper establishes sharp upper bounds on the total irregularity of graphs resulting from various graph operations, enhancing understanding of how irregularity behaves under these transformations.

Contribution

It provides the first comprehensive analysis of total irregularity bounds for multiple graph operations, including join, products, corona, disjunction, and symmetric difference.

Findings

Sharp upper bounds for total irregularity under graph operations

New insights into how graph irregularity changes with operations

General formulas applicable to various graph classes

Abstract

The total irregularity of a graph $G$ 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 $u \in V(G)$. In this paper we give (sharp) upper bounds on the total irregularity of graphs under several graph operations including join, lexicographic product, Cartesian product, strong product, direct product, corona product, disjunction and symmetric difference.