On the extremal cacti with minimum Sombor index

Let H be a graph with edge set EH. The Sombor index and the reduced Sombor index of a graph H are defined as SO (H) = ₔₕ ₄₇d₇ (ₔ) ^₂+d₇ (v) ^{2} and SOₑ₄₃ (H) = ₔₕ ₄₇ (d₇ (ₔ) -₁) ^₂+ (d₇ (v) -1) ^{2}, respectively. Where d₇ (u) and d₇ (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 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 C (n, k) is obtained and the corresponding extremal cacti are characterized when n 4k-2 and k 2. Moreover, the lower bound of the reduced Sombor index of cacti is obtained by similar approach.