Bond Incident Degree Indices of Catacondensed Pentagonal Systems

Yongsheng Rao,Muhammad Unfowan Noor,A Othman Almatroud,Adnan Aslam,Zehui Shao

doi:10.1155/2020/4935760

Abstract

The bond incident degree (BID) indices can be written as a linear combination of the number of edges xi,j with end vertices of degree i and j. We introduce two transformations, namely, linearizing and unbranching, on catacondensed pentagonal systems and show that BID indices are monotone with respect to these transformations. We derive a general expression for calculating the BID indices of any catacondensed pentagonal system with a given number of pentagons, angular pentagons, and branched pentagons. Finally, we characterize the CPSs for which BID indices assume extremal values and compute their BID indices.

Highlights

A pentagonal system is a connected geometric figure obtained by concatenating congruent regular pentagons side to side in a plane in such a way that the figure divides the plane into one infinite region and a number of finite regions, and all internal regions must be congruent regular pentagons
Examples 1 and 2 show that we can transform any catacondensed pentagonal system into linear pentagonal chain by successively applying linearizing and unbranching transformations. e number of steps depends on the number of angular pentagons a2 and number of branched pentagons a3. eorems 1 and 2 show that we can find the exact value of topological indices after applying these transformations
Our results show that once we know the number of angular pentagons a2, the number of branched pentagons a3, and the number of pentagons n in a catacondensed pentagonal system, we can compute its topological indices

Summary

Introduction

A pentagonal system is a connected geometric figure obtained by concatenating congruent regular pentagons side to side in a plane in such a way that the figure divides the plane into one infinite (external) region and a number of finite (internal) regions, and all internal regions must be congruent regular pentagons. Among the degree-based topological descriptors, the most studied are the first and second Zagreb indices [14,15,16,17], the sum-connectivity index [17,18,19], the atom-bond connectivity index [17, 20], the augmented Zagreb index [17, 21, 22], the geometric arithmetic index [23,24,25], and the harmonic index [19, 26, 27]. In case of CPS, we have only vertices of degree 2, 3, and 4; the general BID indices over CHn will be induced by a sequence 􏼈θ22, θ23, θ24, θ33, θ34􏼉 of nonnegative real numbers: BID(G) θ22x22 + θ23x23 + θ24x24 + θ33x33 + θ34x34,.

Linearizing and Unbranching a Catacondensed Pentagonal System

Method to Compute BID Indices of Catacondensed Pentagonal System

Concluding Remarks