Extremal Polygonal Cacti for General Sombor Index.

The Sombor index of a graph $G$ was recently introduced by Gutman from the geometric point of view, defined as $SO(G)=\sum_{uv\in E(G)}\sqrt{d(u)^2+d(v)^2}$, where $d(u)$ is the degree of a vertex $u$. For two real numbers $\alpha$ and $\beta$, the general Sombor index of $G$ is a generalized form of the Sombor index defined as $SO_\alpha(G;\beta)=\sum_{uv\in E(G)}(d(u)^{\beta}+d(v)^{\beta})^{\alpha}$. A $k$-polygonal cactus is a connected graph in which every block is a cycle of length $k$. In this paper, we establish some lower and upper bounds on $SO_\alpha(G;\beta)$ for $k$-polygonal cacti for some particular $\alpha$ and $\beta$, and also characterize the corresponding extremal $k$-polygonal cacti.