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

TL;DR

This paper determines the maximum Sombor index values for unicyclic graphs with a fixed number of vertices and a specified diameter, extending previous work on trees to a broader class of graphs.

Contribution

It introduces the first extremal results for the Sombor index on unicyclic graphs with fixed order and diameter, expanding the scope beyond trees.

Findings

Identified maximum Sombor indices for unicyclic graphs with given parameters.

Extended extremal Sombor index results from trees to unicyclic graphs.

Provides a characterization of extremal unicyclic graphs based on the Sombor index.

Abstract

Sombor index is a novel topological index, which was introduced by Gutman and defined for a graph $G$ as $SO(G)=\sum\limits_{uv\in E(G)}\sqrt{d_{u}^{2}+d_{v}^{2}}$, 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. [H. Chen, W. Li, J. Wang, Extremal values on the Sombor index of trees, MATCH Commun. Math. Comput. Chem. 87 (2022) 23--49] and Li et al. [S. Li, Z. Wang, M. Zhang, On the extremal Sombor index of trees with a given diameter, Appl. Math. Comput. 416 (2022) 126731]. As an extension of results introduces above, we determine the maximum Sombor indices for unicyclic graphs with a fixed order and given diameter.