General sum-connectivity index of unicyclic graphs with given maximum degree

Abstract

For $a\in R$, the general sum-connectivity index ${\chi }_a$ of a graph $G$ is defined as ${\chi }_a\left(G\right)={\sum }_{uv\in E\left(G\right)}{[d_G\left(u\right)+d_G\left(v\right)]}^a$, where $E\left(G\right)$ is the set of edges of $G$, and $d_G\left(u\right)$ and $d_G\left(v\right)$ are the degrees of vertices $u$ and $v$, respectively. Among unicyclic graphs with given number of vertices and maximum degree, we present graphs having the largest and smallest values of ${\chi }_a$, and we state cases which are still open. We also solve one of the open problems on ${\chi }_a$ for trees if $0<a<1$.