Two-Matchings with Respect to the General Sum-Connectivity Index of Trees

Abstract

A vertex-degree-based topological index φ associates a real number to a graph G which is invariant under graph isomorphism. It is defined in terms of the degrees of the vertices of G and plays an important role in chemical graph theory, especially in QSPR/QSAR investigations. A subset of k edges in G with no common vertices is called a k-matching of G, and the number of such subsets is denoted by mG,k. Recently, this number was naturally extended to weighted graphs, where the weight function is induced by the topological index φ. This number was denoted by mkG,φ and called the k-matchings of G with respect to the topological index φ. It turns out that m1G,φ=φG, and so for k≥2, the k-matching numbers mkG,φ can be viewed as kth order topological indices which involve both the topological index φ and the k-matching numbers. In this work, we solve the extremal value problem for the number of 2-matchings with respect to general sum-connectivity indices SCα, over the set Tn of trees with n vertices, when α is a real number in the interval −1,0.