Sombor indices of cacti

Source abstract

<abstract><p>For a graph $G$, the Sombor index $SO(G)$ of $G$ is defined as</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$SO(G) = \sum\limits_{uv\in E(G)}\sqrt{d_{G}(u)^{2}+d_{G}(v)^{2}},$\end{document} </tex-math></disp-formula></p> <p>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 $k(k\geq 2)$ 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 $n\geq 6k-3$, 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 $n \geq 3k$.</p></abstract>