Extremal problems on Sombor indices of unicyclic graphs with a given diameter

Abstract

The Sombor index is a novel topological index, which was introduced by Gutman and defined for a graph G as $SO(G)=\sum \nolimits _{uv\in E(G)}\sqrt{d_{u}^{2}+d_{v}^{2}}$, where $d_{u}=d_{G}(u)$ denotes the degree of vertex u in graph G. Extremal problems on the Sombor index for trees with a given diameter has been considered by Chen et al. (MATCH Commun Math Comput Chem 87:23–49, 2022) and Li et al. (Appl Math Comput 416:126731, 2022). As an extension of the results introduced above, we determine the maximum Sombor indices for unicyclic graphs with a fixed order and given diameter.