The Bounds of Vertex Padmakar–Ivan Index on k-Trees

Shaohui Wang,Bing Wei,Zehui Shao,Jia-Bao Liu

doi:10.3390/math7040324

Abstract

The Padmakar–Ivan ( P I ) index is a distance-based topological index and a molecular structure descriptor, which is the sum of the number of vertices over all edges u v of a graph such that these vertices are not equidistant from u and v. In this paper, we explore the results of P I -indices from trees to recursively clustered trees, the k-trees. Exact sharp upper bounds of PI indices on k-trees are obtained by the recursive relationships, and the corresponding extremal graphs are given. In addition, we determine the P I -values on some classes of k-trees and compare them, and our results extend and enrich some known conclusions.

Highlights

Let G be a simple connected non-oriented graph with vertex set V ( G ) and edge set E( G ).The distance d( x, y) between the vertices x, y ∈ V ( G ) is the minimum length of the paths between x and y in G
Based on the considerable success of Wiener index and Sz index, Khadikar proposed a new distance-based index [3] to be used in the field of nano-technology, that is edge Padmakar–Ivan (PIe ) index, PIe ( G ) = ∑ xy∈E(G) [ne ( x ) + ne (y)], where ne ( x ) denotes the number of edges which are closer to the vertex x than to the vertex y, and ne (y) denotes the number of edges which are closer to the vertex y than to the vertex x, respectively
In [31], Das and Gutman obtained a lower bound on the vertex PI index of a connected graph in terms of numbers of vertices, edges, pendent vertices, and clique number

Summary

Introduction

Let G be a simple connected non-oriented graph with vertex set V ( G ) and edge set E( G ). In [31], Das and Gutman obtained a lower bound on the vertex PI index of a connected graph in terms of numbers of vertices, edges, pendent vertices, and clique number. Provided exact formulas for the vertex PI indices of Kronecker product of a connected graph G and a complete graph. Since the tree is a basic class of graphs in mathematics and chemistry, and these results indicate that either the stars or the paths attain the maximal or minimal bounds for particular chemical indices, a natural question is how about the situations for vertex Padmakar–Ivan index?. Because PI index is a distance-based index and not very easy to calculate, we first consider the bipartite graph G with n vertices. K ∗ ) if c ∈ [1, k +1 ), PI ( Pnk ) ≥ PI ( Tn,c k k ∗ ) if c ∈ [ k +1 , k − 1]

Preliminary

Main Proofs

Conclusions