Tree-like Polyphenyl Chains with Extremal Degree Distance

Let G be a graph with vertex set V(G) and edge set E(G). For any two vertices x and y in V(G), the distance between x and y, denoted by d(x,y), is the length of the shortest path connecting x and y. The degree of a vertex v in G is the number of neighbors of v in G. Numbers reflecting certain structural features of organic molecules that are obtained from the molecular graph are usually called graph invariants or more commonly topological indices. The oldest and most thoroughly examined use of a topological index in chemistry was by Wiener [1] in the study of paraffin boiling points, and the topological index was called Wiener index or Wiener number.The conventional generalization of W for an arbitrary molecular graph is due to Hosoya [2]. The Wiener index of the graph G, is equals to the sum of distances between all pairs of vertices of the respective molecular graph, i.e., ∑ ⊆ = ) ( } , { ) , ( ) ( G V v u v u d G W .