Ordering Trees Having Small General Sum-Connectivity Index

The general sum-connectivity index of a graph G is defined as chi alpha(G) = Sigma(uv is an element of E(G)) (d(u)+d(v))(alpha), where d(u) denotes the degree of vertex u in G, and alpha is a real number. The aim of this paper is twofold. We determine the minimum value of the general sum-connectivity index: (i) for trees of order n >= 3 and diameter d, 2 <= d <= n - 1 and of trees of order n >= 5 having p pendant vertices, 3 <= p <= n - 2 and the corresponding extremal trees for -1 <= alpha < 0 and (ii) for connected multigraphs of order n >= 3 and size m, m >= n - 1 and the corresponding extremal multigraphs for -3 <= alpha < 0. Further, for n sufficiently large and -1 <= alpha < 0, we characterize five n-vertex trees having smallest values of chi alpha.