Sharp Bounds on General Sum-Connectivity Index Based on Lexicographic Product

Give a graph G , let E\left(G\right) and d_G\left(v\right) represent the set of edges and the degree of the vertex v , respectively. For an edge e=uv , the general sum-connectivity index is \chi _\alpha \left(e\right)=(d_G\left(u\right)+d_G(v\left)\right)^\alpha , in which \alpha is any real number. Before taking the product of two simple connected graphs G and H , we first perform four operations of S,R,Q,T on the graph H , denoted as F\left(H\right) , in which F\in \S, R, Q, T\ , then take the lexicographical product of graphs G and F\left(H\right) . The sharp bounds on general sum-connectivity index of graphs for operations based on lexicographic product are given, and these bounds are sharp.