20.500.12556/DiRROS-27405 The ▫$\sigma$▫-irregularity of trees with maximum degree ▫$5$▫ 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. regular graph trees maximum degree [sigma]-irregularity maximal graphs graph measure topological index true false true Elsevier Angleški jezik Ni določen © 2025 The Authors Neznano 2026-02-05 15:17:34 2026-02-05 15:17:34 2026-02-06 03:51:20 0000-00-00 00:00:00 2026 0 0 Žana Kovijanić Vukićević, Goran Popivoda, Jelena Sedlar, Riste Škrekovski, Saša Vujošević; str. 124-136 Vol. 382 Mar. 2026 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2026-03-01 519.17 0166-218X 10.1016/j.dam.2025.11.045 266933507 https://www.sciencedirect.com/science/article/pii/S0166218X25006857 1 6d993fcd-029d-11f1-93b0-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=40086 RAZ_Dimitrov_Darko_2026.pdf RAZ_Dimitrov_Darko_2026.pdf 1 376E48E44FF580DB7A28FE572333A1C8 ab4bf8433fa6d8d7ecad9f3b49a0590f29644aa7a60451252f97bed04d97de63 c5e66b8e-029d-11f1-93b0-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=40087 Rudolfovo – Znanstveno in tehnološko središče Novo mesto 0 0 0