On Extremal Values of the Nk-Degree Distance Index in Trees

The Nk-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 Nk(G)=∑k=1d(G)∑v∈V(G)dk(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 dk(v)={u,v∈V(G)d(u,v)=k}. In this paper, we extend the study of the Nk-index of graphs. We introduced some graph transformations and their impact on the Nk-index of graph and proved that the star graph has the minimum, and the path graph has the maximum Nk-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 Bn,Δ (consisting of a star SΔ+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 Nk-index. Further, we also defined and proved a graph with maximum Nk-index for a given number of n vertices, maximum-degree Δ, and perfect matching among trees. We characterize the starlike trees which minimize the Nk-index and propose a unique tree which minimizes the Nk-index with diameter d and n vertices among trees.