On the Local Value of a Vertex Degree Function Index of Graphs

Source abstract

Let f be a real-valued function defined on the integers in the interval [1, n−1], and let G be a graph with n non-isolated vertices. The concept of the local value of a degree-function index was introduced by Rada in [Local value of a vertex-degree function index of a graph, MATCH Commun. Math. Comput. Chem. 95 (2026) 111–124]. The degree-function index Hf (G) is defined by Hf (G) = ∑︂u∈V (G)f(dG(u)), where dG(u) denotes the degree of the vertex u in G. The local value of Hf at a vertex u, denoted by fG(u), measures the contribution of u to the total value of the index. In the same paper, Rada posed four questions concerning the local value of a degree-function index and the zeroth-order general Randi´c index. In this paper, we provide answers to these questions. We establish an upper bound for the difference between the local values of a vertex-degree function index f at a vertex u in a graph G and in an induced subgraph H of G. Furthermore, we derive an upper bound for the difference between the local values of f at two vertices of a graph G. We also obtain some upper and lower bounds for the zeroth-order general Randi´c local value (Rα)G(u). As applications of these results, we identify the vertices attaining the maximum local value of the zeroth-order general Randi´c index Rα for 0 < α < 1, and the minimum local value for α < 0. Finally, we present some concluding remarks and discuss several possible directions for future research.