On the sum-connectivity index of cacti

For a simple connected graph G=(V,E), X(G)=∑uv∈E1du+dv is its sum-connectivity index, where du denotes the degree of a vertex u. A connected graph G is a cactus if any two of its cycles have at most one common vertex. Let G(n,r) be the set of cacti of order n and with r cycles, ζ(2n,r) the set of cacti of order 2n with a perfect matching and r cycles. In this paper, we give the sharp lower bounds of the sum-connectivity index of cacti among G(n,r) and ζ(2n,r) respectively: (1) if G∈G(n,r), n≥5, then X(G)≥2rn+1+n−2r−1n+r2; (2) if G∈ζ(2n,r), n≥4, then X(G)≥n+r−1n+r+2+1n+r+1+n−r−13+r2, and characterize the corresponding extremal cacti.