The $\sigma$-irregularity of trees with maximum degree $5$

Dimitrov, Darko (author), Kovijanić-Vukićević, Žana (author), Popivoda, Goran (author), Sedlar, Jelena (author), Škrekovski, Riste (author), Vujošević, Saša (author)

The $\sigma$-irregularity, a variant of the well-established Albertson irregularity, is a topological invariant defined for a graph $G=(V,E)$ as $\sigma (G) = \sum_{u,v \in E}(d(u) - d(v))^2$, where $d(u)$ and $d(v)$ denote the degrees of vertices $u$ and $v$, respectively. Recent research has successfully characterized chemical trees with the maximum $\sigma$-irregularity. In this paper, we expand upon this research by establishing several structural properties of maximal trees with prescribed maximum degree $\Delta$. Application of these properties enables us to characterize maximal trees with $\Delta = 5$. We establish that extremal trees contain only vertices of degrees $1$, $2$ and $\Delta$. Moreover, the number of edges with both end-vertices having the degree $2$ or $\Delta$ is very small, so almost all edges have the (second) maximum possible contribution to $\sigma$-irregularity. We believe this property or similar should extend to maximal trees for any value of $\Delta$, so this is an interesting direction for further research.

• Language: English
• Keywords: regular graph, trees, maximum degree, [sigma]-irregularity, maximal graphs, graph measure, topological index
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.03.2026
• Year: 2026
• Number of pages: Str. 124-136
• Numbering: Vol. 382
• PID: 20.500.12556/RUL-179011
• UDC: 519.17
• ISSN on article: 0166-218X
• DOI: 10.1016/j.dam.2025.11.045
• COBISS.SI-ID: 266933507
• Publication date in RUL: 03.02.2026

Record is a part of a journal

• Title: Discrete applied mathematics
• Shortened title: Discrete appl. math.
• Publisher: Elsevier
• ISSN: 0166-218X
• COBISS.SI-ID: 25342464

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