Sombor indices of cacti

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>