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

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

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.

• Jezik: Angleški jezik
• Ključne besede: regular graph, trees, maximum degree, [sigma]-irregularity, maximal graphs, graph measure, topological index
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FMF - Fakulteta za matematiko in fiziko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Datum objave: 01.03.2026
• Leto izida: 2026
• Št. strani: Str. 124-136
• Številčenje: Vol. 382
• PID: 20.500.12556/RUL-179011
• UDK: 519.17
• ISSN pri članku: 0166-218X
• DOI: 10.1016/j.dam.2025.11.045
• COBISS.SI-ID: 266933507
• Datum objave v RUL: 03.02.2026

Gradivo je del revije

• Naslov: Discrete applied mathematics
• Skrajšan naslov: Discrete appl. math.
• Založnik: Elsevier
• ISSN: 0166-218X
• COBISS.SI-ID: 25342464

Licence

• Licenca: CC BY-NC-ND 4.0, Creative Commons Priznanje avtorstva-Nekomercialno-Brez predelav 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by-nc-nd/4.0/deed.sl
• Opis: Najbolj omejujoča licenca Creative Commons. Uporabniki lahko prenesejo in delijo delo v nekomercialne namene in ga ne smejo uporabiti za nobene druge namene.