On the general Sombor index of unicyclic graphs with a given diameter

Abstract

The general Sombor index of $G$ is defined as $S{O}_{\alpha }(G)={\sum }_{uv\in G}{\left(d_G^2(u)+d_G^2(v)\right)}^{\alpha }$. For $0<\alpha <1$, we have the upper bound of $S{O}_{\alpha }(G)$ on unicyclic graphs with a fixed diameter, and the extremal graph is also characterized.

12 pages, 5 figures