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Some Results on $\Sigma_{t}$-irregularity by Filipovski, Slobodan; Dimitrov, Darko; Knor, Martin; Škrekovski, Riste is a scholarly article available to read on EtoBox.

What is Some Results on $\Sigma_{t}$-irregularity about?

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 $\sigma_{t}$-ir

Author

Filipovski, Slobodan; Dimitrov, Darko; Knor, Martin; Škrekovski, Riste

Published

2024

Language

EN