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) regular graphtreesmaximum degree[sigma]-irregularitymaximal graphsgraph measuretopological indexThe ▫$\sigma$▫-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.Elsevier20262026-02-05 15:17:34Neznano27405UDK: 519.17ISSN pri članku: 0166-218XDOI: 10.1016/j.dam.2025.11.045COBISS_ID: 266933507sl© 2025 The Authors