On the extremal cacti with minimum Sombor index

Let $ H $ be a graph with edge set $ E_H $. The Sombor index and the reduced Sombor index of a graph $ H $ are defined as $ SO(H) = \sum\limits_{uv\in E_H}\sqrt{d_{H}(u)^{2}+d_{H}(v)^{2}} $ and $ SO_{red}(H) = \sum\limits_{uv\in E_H}\sqrt{(d_{H}(u)-1)^{2}+(d_{H}(v)-1)^{2}} $, respectively. Where $ d_{H}(u) $ and $ d_{H}(v) $ are the degrees of the vertices $ u $ and $ v $ in $ H $, respectively. A cactus is a connected graph in which any two cycles have at most one common vertex. Let $ \mathcal{C}(n, k) $ be the class of cacti of order $ n $ with $ k $ cycles. In this paper, the lower bound for the Sombor index of the cacti in $ \mathcal{C}(n, k) $ is obtained and the corresponding extremal cacti are characterized when $ n\geq 4k-2 $ and $ k\geq 2 $. Moreover, the lower bound of the reduced Sombor index of cacti is obtained by similar approach.