On the sum-connectivity index of cacti

For a simple connected graph G = (V, E), X(G) = Sigma uv is an element of E 1/root d(u)+d(v) is its sum-connectivity index, where d(u) 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, zeta (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 zeta (2n, r) respectively: (1) if G is an element of g(n, r), n >= 5, then X(G) >= 2r/root n+1 + n-2r-1/root n + r/2; (2) if G is an element of zeta (2n, r), n >= 4, then X(G) >= n+r-1/root n+r+2 + 1/root n+r+1 + n-r-1/root 3 + r/2, and characterize the corresponding extremal cacti. (C) 2011 Elsevier Ltd. All rights reserved.