Sombor indices of cacti

For a graph $G$, the Sombor index $SO(G)$ of $G$ is defined as

$$SO(G)=\sum _{uv\in E(G)}\sqrt{d_G(u{)}^2+d_G(v{)}^2},$$

where $d_G(u)$ is the degree of the vertex $u$ in $G$. A cactus is a connected graph in which each block is either an edge or a cycle. Let $\mathcal{G}(n,k)$ be the set of cacti of order $n$ and with $k$ cycles. Obviously, $\mathcal{G}(n,0)$ is the set of all trees and $\mathcal{G}(n,1)$ is the set of all unicyclic graphs, then the cacti of order $n$ and with $k(k\ge 2)$ cycles is a generalization of cycle number $k$. In this paper, we establish a sharp upper bound for the Sombor index of a cactus in $\mathcal{G}(n,k)$ and characterize the corresponding extremal graphs. In addition, for the case when $n\ge 6k-3$, we give a sharp lower bound for the Sombor index of a cactus in $\mathcal{G}(n,k)$ and characterize the corresponding extremal graphs as well. We also propose a conjecture about the minimum value of sombor index among $\mathcal{G}(n,k)$ when $n\ge 3k$.