Sombor indices of cacti

For a graph G, the Sombor index SO (G) of G is defined as document SO (G) = ₔₕ ₄ (₆) d₆ (ₔ) ^₂+d₆ (v) ^{2}, document where d₆ (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 G (n, k) be the set of cacti of order n and with k cycles. Obviously, G (n, 0) is the set of all trees and G (n, 1) is the set of all unicyclic graphs, then the cacti of order n and with k (k 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 G (n, k) and characterize the corresponding extremal graphs. In addition, for the case when n 6k-3, we give a sharp lower bound for the Sombor index of a cactus in G (n, k) and characterize the corresponding extremal graphs as well. We also propose a conjecture about the minimum value of sombor index among G (n, k) when n 3k.