Asymmetric Circular Graph with Hosoya Index and Negative Continued Fractions

TL;DR

This paper extends the calculation of the Hosoya index to asymmetric circular graphs with non-uniform ring structures using negative continued fractions, linking graph indices to advanced continued fraction concepts.

Contribution

It introduces a method to compute the Hosoya index for asymmetric circular graphs via negative continued fractions, generalizing previous work on caterpillar and caterpillar-bond graphs.

Findings

Hosoya index of asymmetric circular graphs can be derived from negative continued fractions.

Established a relation between radial graphs and multidimensional continued fractions.

Extended the framework for calculating graph indices beyond simple structures.

Abstract

It has been known that the Hosoya index of caterpillar graph can be calculated as the numerator of the simple continued fraction. Recently, the author \cite{Komatsu2020} introduces a more general graph called caterpillar-bond graph and shows that its Hosoya index can be calculated as the numerator of the general continued fraction. In this paper, we show how the Hosoya index of the graph with non-uniform ring structure can be calculated from the negative continued fraction. We also give the relation between some radial graphs and multidimensional continued fractions in the sense of the Hosoya index.