Extremal Graphs for Topological Index Defined by a Degree-Based Edge-Weight Function

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 T I f (G) of G defined by a degree-based edge-weight function f (x, y) is given asy) satisfies some of following properties: f 1 > 0, f 11 > 0, f 12 ≥ 0, f 111 ≥ 0 and for any, we obtain some upper bounds and lower bounds for the topological index T I 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 T I f (G).