On Extremal Values of the N k -Degree Distance Index in Trees

The N k -index ( k -distance degree index) of a connected graph G was first introduced by Naji and Soner as a generalization of the distance degree concept, as N k ( G ) = ∑ k = 1 d ( G ) ∑ v ∈ V ( G ) d k ( v ) k , where the distance between u and v in G is denoted by d ( u , v ) , the diameter of a graph G is denoted by d ( G ) , and the degree of a vertex v at distance k is denoted by d k ( v ) = { u , v ∈ V ( G ) d ( u , v ) = k } . In this paper, we extend the study of the N k -index of graphs. We introduced some graph transformations and their impact on the N k -index of graph and proved that the star graph has the minimum, and the path graph has the maximum N k -index among the set of all trees on n vertices. We also show that among all trees with fixed maximum-degree Δ , the broom graph B n , Δ (consisting of a star S Δ + 1 and a pendant path of length n − Δ − 1 attached to any arbitrary pendant path of star) is a unique tree which maximizes the N k -index. Further, we also defined and proved a graph with maximum N k -index for a given number of n vertices, maximum-degree Δ , and perfect matching among trees. We characterize the starlike trees which minimize the N k -index and propose a unique tree which minimizes the N k -index with diameter d and n vertices among trees.