Some results on $\sigma_{t}$-irregularity

TL;DR

This paper explores the sigma total index as a measure of graph irregularity, characterizes graphs with maximum irregularity, and establishes bounds and relationships with other graph invariants.

Contribution

It provides a new characterization of maximum sigma total index graphs, relates sigma to degree variance and graph energy, and introduces new bounds and conjectures.

Findings

Maximum sigma irregularity graphs are split graphs and specific complete bipartite graphs.

Sigma total index equals the number of vertices squared times the degree variance.

New bounds and relationships between sigma, Laplacian eigenvalues, and graph energy are established.

Abstract

The $\sigma_{t}$-irregularity (or sigma total index) is a graph invariant which is defined as $\sigma_{t}(G)=\sum_{\{u,v\}\subseteq V(G)}(d(u)-d(v))^{2},$ where $d(z)$ denotes the degree of $z$. This irregularity measure was proposed by R\' {e}ti [Appl. Math. Comput. 344-345 (2019) 107-115], and recently rediscovered by Dimitrov and Stevanovi\'c [Appl. Math. Comput. 441 (2023) 127709]. In this paper we remark that $\sigma_{t}(G)=n^{2}\cdot Var(G)$, where $Var(G)$ is the degree variance of the graph. Based on this observation, we characterize irregular graphs with maximum $\sigma_{t}$-irregularity. We show that among all connected graphs on $n$ vertices, the split graphs $S_{\lceil\frac{n}{4}\rceil, \lfloor\frac{3n}{4}\rfloor }$ and $S_{\lfloor\frac{n}{4}\rfloor, \lceil\frac{3n}{4}\rceil }$ have the maximum $\sigma_{t}$-irregularity, and among all complete bipartite graphs on $n$ vertices, either the complete bipartite graph $K_{\lfloor\frac{n}{4}(2-\sqrt{2})\rfloor, \lceil\frac{n}{4}(2+\sqrt{2})\rceil }$ or $K_{\lceil\frac{n}{4}(2-\sqrt{2})\rceil, \lfloor\frac{n}{4}(2+\sqrt{2})\rfloor }$ has the maximum sigma total index. Moreover, various upper and lower bounds for $\sigma_{t}$-irregularity are provided; in this direction we give a relation between the graph energy $\mathcal{E}(G)$ and sigma total index $\sigma_{t}(G)$ and give another proof of two results by Dimitrov and Stevanovi\'c. Applying Fiedler's characterization of the largest and the second smallest Laplacian eigenvalue of the graph, we also establish new relationships between $\sigma_{t}$ and $\sigma$. We conclude the paper with two conjectures.