Irregularity of expansions and Pell graphs

TL;DR

This paper investigates the irregularity of graphs, focusing on expansions and Pell graphs, by deriving formulas for irregularity and edge imbalance distributions, and analyzing their properties.

Contribution

It provides explicit formulas for the irregularity of expansions and Pell graphs, and studies the imbalance distribution of edges in Pell graphs.

Findings

Derived formulas for irregularity of expansions.

Analyzed edge imbalance distribution in Pell graphs.

Expressed the number of edges with fixed imbalance in Pell graphs.

Abstract

For a graph $G$ the imbalance of an edge $uv$ of $G$ is $|deg_G(u)-deg_G(v)|$. Irregularity of a graph $G$ is defined as the sum of imbalances over all edges of $G$. In this paper we consider expansions and Pell graphs. If $H$ is an expansion of $G$ with respect to the sets $V_1$ and $V_2$, we express the irregularity of $H$ using $G, V_1$ and $V_2$. For Pell graphs the imbalance of their edges is studied. The number of edges of a Pell graph with a fixed imbalance $k$ is expressed. Using these results the irregularity and the $\sigma$-index of Pell graphs are given.