Abstract
For a simple graph \(G\), \(d_u\) denotes the degree of a vertex \(u\) in \(G\). Let \(f(x,y)\) be a symmetric real function in two variables, and define the weight \(w(e)\) of an edge \(e=uv\) of \(G\) by \(w(e)=f(d_u, d_v).\) Then the topological index
\(TI_f(G)\) of \(G\) defined by a degree-based edge-weight function \(f(x,y)\) is given as \(TI_f(G)=\sum_{uv\in E(G)}f(d_u,d_v).\) Let \(f_1(x,y)=f(x+1,y)-f(x,y)\), \(f_2(x,y)=f(x,y+1)-f(x,y)\), \(f_{11}=(f_1)_1\), \(f_{12}=(f_1)_2\)
and \(f_{111}=(f_{11})_1\). If \(f(x,y)\) satisfies some of following properties: \(f_1>0,f_{11}>0,f_{12}\geq 0,f_{111}\geq 0\) and for any \(x_1+y_1=x_2+y_2\) with \(|x_1-y_1|>|x_2-y_2|\), \(f(x_1,y_1)>f(x_2,y_2)\), we obtain some upper
bounds and lower bounds for the topological index \(TI_f(G)\) and give some graphs of given order and size achieving the bounds. For graphs with small size, we characterize the graphs with maximal and minimal values of the index \(TI_f(G)\).