Asymmetric circular graph with Hosoya index and negative continued fractions

It has been known that the Hosoya index of caterpillar graph can be calculated as the numerator of the simple continued fraction. Recently in [MATCH Commun. Math. Comput. Chem. 2020, 84 (2), 399-428], the author 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.

On the Hosoya Indices of Bicyclic Graphs with Small Diameter

Let G be a graph. The Hosoya index of G , denoted by z G , is defined as the total number of its matchings. The computation of z G is NP-Complete. Wagner and Gutman pointed out that it is difficult to obtain results of the maximum Hosoya index among tree-like graphs with given diameter. In this paper, we focus on the problem, and a sharp bound of Hosoya indices of all bicyclic graphs with diameter of 3 is determined.

Matching polynomials for some nanostar dendrimers

Dendrimers are highly branched monodisperse, macromolecules and are considered in nanotechnology with a variety of suitable applications. In this paper, the matching polynomial and some results of the matchings for three classes of nanostar dendrimers are obtained. Furthermore, we express the recursive formulas of the Hosoya index for these structures of dendrimers by their matching polynomials.

Linear Algorithms for the Hosoya Index and Hosoya Matrix of a Tree

The Hosoya index of a graph is defined as the total number of its independent edge sets. This index is an important example of topological indices, a molecular-graph based structure descriptor that is of significant interest in combinatorial chemistry. The Hosoya index inspires the introduction of a matrix associated with a molecular acyclic graph called the Hosoya matrix. We propose a simple linear-time algorithm, which does not require pre-processing, to compute the Hosoya index of an arbitrary tree. A similar approach allows us to show that the Hosoya matrix can be computed in constant time per entry of the matrix.

Hosoya properties of the commuting graph associated with the group of symmetries

A vast amount of information about distance based graph invariants is contained in the Hosoya polynomial. Such an information is helpful to determine well-known distance based molecular descriptors. The Hosoya index or Z-index of a graph G is the total number of its matching. The Hosoya index is a prominent example of topological indices, which are of great interest in combinatorial chemistry, and later on it applies to address several chemical properties in molecular structures. In this article, we investigate Hosoya properties (Hosoya polynomial, reciprocal Hosoya polynomial and Hosoya index) of the commuting graph associated with an algebraic structure developed by the symmetries of regular molecular gones (constructed by atoms with regular atomic-bonding).

Erratum: Liu, W.; Ban, J.; Feng, L.; Cheng, T.; Emmert-Streib, F.; Dehmer, M. The Maximum Hosoya Index of Unicyclic Graphs with Diameter at Most Four. Symmetry 2019, 11, 1034

The authors wish to make the following corrections to their paper [...]