On the Wiener polarity index of trees with maximum degree or given number of leaves

Abstract

a b s t r a c tThe Wiener polarity index W P (G) of a graph G = (V , E) is the number of unordered pairs of vertices {u, v} of G such that the distance d G (u, v) = 3. In this paper, the minimum (resp. maximum) Wiener polarity index of trees with n vertices and maximum degree ∆ are given, and the corresponding extremal trees are determined, where 2 ≤ ∆ ≤ n − 1. Moreover, the trees minimizing W P (T ) among all trees T of order n and k leaves are characterized, where 2 ≤ k ≤ n − 1.

On Minimum Wiener Polarity Index of Unicyclic Graphs with Prescribed Maximum Degree

“…After joining 1 and 2 vertices, each by an edge, to the two endpoints of a path ,, respectively, we obtain a new graph and denote it by 1 , 2 [11]. For example, the path on two vertices may be regarded as 2 0,0 , and the star with ≥ 4 may be regarded as…”

The Alkanes with Maximum Wiener Polarity Index

“…The extremal trees with respect to in the collection of all n ‐vertex trees having fixed diameter were identified in the paper . The problem of finding trees having maximum and minimum values among all n ‐vertex trees with fixed maximum degree (and with fixed pendant vertices) was solved in the references ,. Recently, the first three minimum values of molecular trees were determined in ,.…”

“…[28] The problem of finding trees having maximum and minimum W p values among all n-vertex trees with fixed maximum degree (and with fixed pendant vertices) was solved in the references. [29,30] Recently, the first three minimum W p values of molecular trees were determined in. [31,32] Some results concerning Wiener polarity index of graphs, other than trees, can be found in the papers.…”

On the generalized Wiener polarity index for some classes of graphs

“…The Wiener polarity index of a graph G = (V (G), E(G)) is defined as W P (G) = |{{u, v} : d(u, v) = 3; u, v ∈ V (G)}| which is a number of unordered pairs of vertices {u, v} of G such that d(u, v) = 3. Authors of [4,5,7,13] studied this index for trees with different parameters such that number of pendant vertices, diameter or maximum degree. Additionally, in [12] there are described algorithms for counting W k (T ) for trees.…”