Some results on $\sigma_t$-irregularity

Filipovski, Slobodan (author), Dimitrov, Darko (author), Knor, Martin (author), Škrekovski, Riste (author)

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 Réti in 2019, and recently rediscovered by Dimitrov and Stevanović in 2023. In this paper we remark that $\sigma_t(G) = n^2\operatorname{Var}(G)$, where $\operatorname{Var}(G)$ is the degree variance of the graph. We show that among all complete bipartite graphs on $n$ vertices, either the complete bipartite graph $K_{\lfloor{n \over 4}(2-\sqrt{2})\rfloor, \lceil {n \over 4}(2+\sqrt{2})\rceil}$ or $K_{\lceil{n \over 4}(2-\sqrt{2})\rceil, \lfloor{n \over 4}(2+\sqrt{2})\rfloor}$ 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 new proofs of two results by Dimitrov and Stevanović. 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.

• Language: English
• Keywords: irregularity, total irregularity, energy of graphs, Laplacian eigenvalues
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2026
• Number of pages: Str. 1-12
• Numbering: Vol. 26, no. 4, art. P4.01
• PID: 20.500.12556/RUL-185661
• UDC: 519.17
• ISSN on article: 1855-3974
• DOI: 10.26493/1855-3974.3268.60f
• COBISS.SI-ID: 254097667
• Publication date in RUL: 17.08.2026

Record is a part of a journal

• Title: Ars mathematica contemporanea
• Publisher: Univerza na Primorskem
• ISSN: 1855-3974
• COBISS.SI-ID: 239051776

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: nepravilnost, iregularnost, popolna nepravilnost, energija grafov, Laplaceove lastne vrednosti