Abstract:
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.