Fibonacci numbers and Lucas numbers in graphs

The Hosoya index and the Merrifield-Simmons index are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure. In recent years, quite a lot of work has been done on the extremal problem for these two indices, i.e., the problem of determining the graphs within certain prescribed classes that maximize or minimize the index value. This survey collects and classifies these results, and also provides some useful auxiliary results, tools and techniques that are frequently used in the study of this type of problem.Keywords Hosoya index · Merrifield-Simmons index · Extremal problem · Molecular graphs Mathematics Subject Classification (2000) 92E10 · 05C35 · 05C70 History and Chemical BackgroundIn 1971 the Japanese chemist Haruo Hosoya introduced a molecular-graph based structure descriptor [35], which he named topological index and denoted by Z. He showed that certain physico-chemical properties of alkanes (= saturated hydrocarbons)-in particular, their boiling points-are well correlated with Z. He defined the quantity Z in the following manner.Let G be a (molecular) graph. Denote by m(G, k) the number of ways in which k mutually independent edges can be selected in G. By definition, m(G, 0) = 1 for all graphs, and

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“…Very similar results were obtained by Startek et al [94], who studied graphs with two elementary (disjoint) cycles.…”

Section: Its Hosoya Index Issupporting

confidence: 75%

Maxima and Minima of the Hosoya Index and the Merrifield-Simmons Index

2010

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“…A wealth of theoretical results on these two parameters has been obtained in recent years, in particular regarding trees and tree-like structures (such as unicyclic graphs [3][4][5] or bicyclic graphs [6,7]). Upper and lower bounds are known under various restrictions, such as diameter [8], number of leaves [9], or number of cut edges [10].…”

Section: Introductionmentioning

confidence: 99%

On the number of independent subsets in trees with restricted degrees

Andriantiana¹,

Wagner²

2011

Mathematical and Computer Modelling

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a b s t r a c tWe study the number of independent vertex subsets (known as the Merrifield-Simmons index in mathematical chemistry) and the number of independent edge subsets (called the Hosoya index) for trees whose vertex degrees are restricted to 1 or d (for some d ≥ 3), a natural restriction in the chemical context. We find that the minimum of the Merrifield-Simmons index and the maximum of the Hosoya index are both attained for path-like trees; furthermore, one obtains the second-smallest value of the Merrifield-Simmons index and the second-largest value of the Hosoya index for generalized tripods. Analogous results are also found for a closely related parameter, the graph energy, which also plays an important rôle in mathematical chemistry.

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“…The Golden Number is closely related to the Fibonacci sequence which is defined recursively by F n = F n−1 + F n−2 for n ≥ 2 with the initial terms F 0 = F 1 = 1. Some new interesting properties and applications of the Fibonacci sequence were studied in [34,35].…”

Section: The Exact Results Of the Diagonalizationmentioning

confidence: 99%

The Exact Results in the One-dimensional Attractive Hubbard Model

Jakubczyk¹

2014

Acta Phys. Pol. B

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The one-dimensional attractive Hubbard model (U 0) is discussed, assuming periodic boundary conditions and the half-filling case. The considered chains have N nodes, the same number of electrons, where N − 1 of them have the same spin projection. The exact diagonalization was performed for any number N of atoms. The eigenvectors and eigenvalues in some cases are constructed based on the Golden Number.

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Fibonacci numbers and Lucas numbers in graphs

Cited by 16 publications

References 3 publications

Maxima and Minima of the Hosoya Index and the Merrifield-Simmons Index

Maxima and Minima of the Hosoya Index and the Merrifield-Simmons Index

On the number of independent subsets in trees with restricted degrees

The Exact Results in the One-dimensional Attractive Hubbard Model

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