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.