On Extremal Values of the Nk-Degree Distance Index in Trees - Zia Ullah Khan; Quaid Iqbal (Mathematics)
The <semantics>Nk</semantics>-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 <semantics>Nk(G)=<mstyle displaystyle="true"><munderover>∑k=1d(G)</munderover></mstyle><mfenced separators="" open="(" close=")"><mstyle displaystyle="true"><munder>∑v∈V(G)</munder></mstyle>dk(v)</mfenced>k</semantics>, where the distance between u and v in G is denoted by <semantics>d(u,v)</semantics>, the diameter of a graph G is denoted by <semantics>d(G)</semantics>, and the degree of a vertex v at distance k is denoted by <semantics>dk(v)={u,v∈V(G)<mspace width="0.166667em"></mspace><mspace width="4pt"></mspace>d(u,v)=k}</semantics>. In this paper, we extend the study of the <semantics>Nk</semantics>-index of graphs. We introduced some graph transformations and their impact on the <semantics>Nk</semantics>-index of graph and proved that the star graph has the minimum, and the path graph has the maximum <semantics>Nk</semantics>-index among the set of all trees on n vertices. We also show that among all trees with fixed maximum-degree <semantics>Δ</semantics>, the broom graph <semantics>Bn,Δ</semantics> (consisting of a star <semantics>SΔ+1</semantics> and a pendant path of length <semantics>n−Δ−1</semantics> attached to any arbitrary pendant path of star) is a unique tree which maximizes the <semantics>Nk</semantics>-index. Further, we also defined and proved a graph with maximum <semantics>Nk</semantics>-index for a given number of n vertices, maximum-degree <semantics>Δ</semantics>, and perfect matching among trees. We characterize the starlike trees which minimize the <semantics>Nk</semantics>-index and propose a unique tree which minimizes the <semantics>Nk</semantics>-index with diameter d and n vertices among trees.