Some Topological Invariants of Generalized M\"obius Ladder

TL;DR

This paper derives a closed-form expression for the Hosoya polynomial of generalized M"obius ladders and demonstrates how to extract various distance-based graph invariants relevant to molecular property prediction.

Contribution

It provides the first general closed-form formula for the Hosoya polynomial of generalized M"obius ladders and shows how to compute multiple graph invariants from it.

Findings

Closed-form Hosoya polynomial for generalized M"obius ladder $M(m,n)$ for arbitrary $m$ and $n=3$

Recovery of Wiener, hyper Wiener, Tratch-Stankevitch-Zefirov, and Harary indices from the polynomial

Enhanced understanding of topological invariants in molecular graph theory

Abstract

The Hosoya polynomial of a graph $G$ was introduced by H. Hosoya in 1988 as a counting polynomial, which actually counts the number of distances of paths of different lengths in $G$. The most interesting application of the Hosoya polynomial is that almost all distance-based graph invariants, which are used to predict physical, chemical and pharmacological properties of organic molecules, can be recovered from it. In this article we give the general closed form of the Hosoya polynomial of the generalized M\"obius ladder $M(m,n)$ for arbitrary $m$ and for $n=3$. Moreover, we recover Wiener, hyper Wiener, Tratch-Stankevitch-Zefirov, and Harary indices from it.