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.20262026-02-03 09:23:24Članek v reviji179011UDK: 519.17ISSN pri članku: 0166-218XDOI: 10.1016/j.dam.2025.11.045COBISS_ID: 266933507sl