Title:
Extremal Graphs for Topological Index Defined by a Degree-Based Edge-Weight Function
Authors:
Zhoukun Hu, Luyi Li, Xueliang Li, Danni Peng
Volume
88
Issue
3
Year
2022
Pages
505-520
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 $$TI_f(G)$$.