On the minimal energy of trees with a given diameter

Yan, Weigen; Ye, Luzhen

doi:10.1016/j.aml.2004.11.001

Cited by 68 publications

(26 citation statements)

References 10 publications

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“…This theorem occurs in several papers by different authors (for the Merrifield-Simmons index in [54,62,87] and for the Hosoya index in [5,84,111]), partly as auxiliary result (as in [54]) or corollary to a more general theorem (see for instance [111], where minimality of the energy is established and minimality of the Hosoya index is stated as a byproduct).…”

Section: Trees With Fixed Diametermentioning

confidence: 88%

“…Theorem 4 (See [5,54,62,84,87,111]) Among all trees with n vertices and diameter d, the broom is the tree that maximizes the Merrifield-Simmons index and minimizes the Hosoya index. This theorem occurs in several papers by different authors (for the Merrifield-Simmons index in [54,62,87] and for the Hosoya index in [5,84,111]), partly as auxiliary result (as in [54]) or corollary to a more general theorem (see for instance [111], where minimality of the energy is established and minimality of the Hosoya index is stated as a byproduct).…”

Section: Trees With Fixed Diametermentioning

confidence: 99%

“…In this case, also Z(T 1 ) ≥ Z(T 2 ). This is also the reason for the fact that in many known instances, the tree within a particular class that maximizes/minimizes the energy is also the one that maximizes/minimizes the Hosoya index [55,81,110,111]. This has even been proven for some non-tree classes, see for instance [50].…”

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confidence: 94%

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

2010

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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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Section: Trees With Fixed Diametermentioning

confidence: 88%

Section: Trees With Fixed Diametermentioning

confidence: 99%

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

2010

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“…Further, within the same class of trees, the broom has the minimum Wiener index and Laplacianenergy like invariant [21]. In [24], it was demonstrated that the broom has minimum energy among trees with the fixed diameter.…”

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confidence: 99%

Distance spectral radius of trees with fixed maximum degree

Stevanović¹,

Ilić²

2010

ELA

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Abstract. Distance energy is a newly introduced molecular graph-based analog of the total π-electron energy, and it is defined as the sum of the absolute eigenvalues of the molecular distance matrix. For trees and unicyclic graphs, distance energy is equal to the doubled value of the distance spectral radius. In this paper, we introduce a general transformation that increases the distance spectral radius and provide an alternative proof that the path Pn has the maximal distance spectral radius among trees on n vertices. Among the trees with a fixed maximum degree ∆, we prove that the broom B n,∆ (consisting of a star S ∆+1 and a path of length n − ∆ − 1 attached to an arbitrary pendent vertex of the star) is the unique tree that maximizes the distance spectral radius, and conjecture the structure of a tree which minimizes the distance spectral radius. As a first step towards this conjecture, we characterize the starlike trees with the minimum distance spectral radius.

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“…[4][5][6][7][8][9][10][11][12][13][14][15]). Note that the concept of a matching is a classical one in graph theory (see ref.…”

Section: Introductionmentioning

confidence: 99%