Local Value of a Vertex-Degree Function Index of a Graph

Let n be a positive integer and f a real function defined on integers in the interval [1, n -1].Given a graph G with vertex set V and n non-isolated vertices, the degree-function index of G is defined as H f (G) = uV f (du).It is our main objective in this paper to introduce the local value of a degree-function index H f of a graph G at a vertex u, which we denote by fG (u).Intutively, fG (u) measures the contribution of vertex u in H f (G).In this paper we initiate the study of its mathematical properties and address the problem of vertices with extremal local values in the zeroth-order general Randi index.In particular, for the first Zagreb index and the forgotten index, the problem of vertices with extremal local values is completely solved.