Base Polynomials for Schultz Invariants of Linear Phenylenes

TL;DR

This paper introduces polynomial bases for calculating Schultz invariants of linear phenylenes, enabling efficient computation of Schultz-related molecular descriptors for these chemical graph structures.

Contribution

It provides polynomial bases representing path lengths among specific vertex degrees in linear phenylenes, facilitating calculation of Schultz invariants.

Findings

Derived polynomials for path lengths in $L_{n}$ and $L'_{n}$

Formulas for Schultz polynomial and indices based on these polynomials

Enhanced methods for computing molecular descriptors in chemical graph theory

Abstract

Let $L_{n}$ be the molecular graph of linear $[n]$phenylene, and $L'_{n}$ the graph obtained by attaching 4-membered cycles to terminal hexagons of $L_{n-1}$. Thus, $L'_{n}$ is the molecular graph of the $\alpha,\omega$ - dicyclobutadieno derivative of $[n-1]$phenylene, containing $n-1$ hexagons and $n$ squares. In this paper we give polynomials which serve as bases for Schultz invariants. Actually, we represent lengths of paths among vertices of degrees 2-2, 2-3, and 3-3 of $L_{n}$ and $L'_{n}$ in terms of polynomials, which are used to find Schultz polynomial, modified Schultz polynomial, Schultz index, and modified Schultz index.