The General Sum-connectivity Index Of Graphs

The general sum-connectivity index of graphs is a new topological index for descripting the molecular structure,and plays an important role in studying the stucture,physical and chemical properties of molecules.The general sum-connectivity x_?(G) of a graph G is defined as the sum of the weights (d(u)+d(v))~? over all edges uv of G,where d(u) denotes the degree of a vertex u of G and ? is a real number.Since ? is an arbitrary number,it is difficult to study the general sum-connectivity index of a graph.Therefore,based on the analysis and summary of the research status at home and abroad,the paper considers the general sum-connectivity index of a certain interval and a certain special graph,such as trees with given maximum matching number and cacti with given number of cycles,pendent vertices and with a perfect matching respectively.Firstly,we consider the trees with given maximum matching number and based on the degree of a certain vertex in the tree,we give the minimum general sum-connectivity indices for ??1 and ??-2 by the mathematical induction and contradiction.Secondly,we study the minimum harmonic index of cacti with given number of cycles by induction on n+r for the two cases of the minimum degree of cacti is 1 and at least 2.For the cactus with given number of pendant vertices,we suppose that there is a counterexample G*,whose number of vertices,pendant vertices and the harmonic index are as small as possible.And we obtain the minimum harmonic index and the corresponding extremal graph.Finally,we determine the minimum harmonic index of cacti with a perfect matching and number of cycles by using mathematical induction method.