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 $\alpha$-Sombor index and general Sombor index of $G$ are two generalized forms of the Sombor index defined as $SO_\alpha(G)=\sum_{uv\in E(G)}(d(u)^{\alpha}+d(v)^{\alpha})^{1/\alpha}$ and $SO_\alpha(G;\beta)=\sum_{uv\in E(G)}(d(u)^{\alpha}+d(v)^{\alpha})^{\beta}$, respectively. A $k$-polygonal cactus is a connected graph in which every block is a cycle of length $k$. In this paper, we establish a lower bound on $\alpha$-Sombor index for $k$-polygonal cacti and show that the bound is attained only by chemical $k$-polygonal cacti. The extremal $k$-polygonal cacti for $SO_\alpha(G;\beta)$ with some particular $\alpha$ and $\beta$ are also considered.