Abstract
The Wiener polarity index of a graphG, usually denoted byWpG, is defined as the number of unordered pairs of those vertices ofGthat are at distance 3. A vertex of a tree with degree at least 3 is called a branching vertex. A segment of a treeTis a nontrivial pathSwhose end-vertices have degrees different from 2 inTand every other vertex (if exists) ofShas degree 2 inT. In this note, the best possible sharp lower bounds on the Wiener polarity indexWpare derived for the trees of fixed order and with a given number of branching vertices or segments, and all the trees attaining this lower bound are characterized.
Highlights
A topological index is a numerical quantity calculated from a graph, which remains unchanged under graph isomorphism [1]
E Wiener polarity index Wp is one of the oldest topological indices, which was proposed in 1947 by the chemist Harold Wiener [5], for predicting the boiling points of paraffins. e index Wp for a graph G is defined as the number of unordered pairs of those vertices of G that are at distance 3
Let G be a graph with the vertex set V(G) and the set of edges E(G). e degree of a vertex u ∈ V(G) is denoted by du(G). e number of vertices in a graph is known as its order
Summary
A topological index is a numerical quantity calculated from a graph, which remains unchanged under graph isomorphism [1]. For fixed integers n and s, denote by STn,s and Tn,b the classes of all n-vertex trees with s segments and b branching vertices, respectively, where 1 ≤ s ≤ n − 1 and 1 ≤ b ≤ (n/2) − 1. Denote by Sns ∈ STn,s the starlike tree with s − 1 pendent paths of length 1 (see Figure 1).
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