On kites, comets, and stars. Sums of eigenvector coefficients in (molecular) graphs.

Research output: Journal contributionsJournal articlesResearchpeer-review

Standard

On kites, comets, and stars. Sums of eigenvector coefficients in (molecular) graphs. / Rücker, Christoph; Rücker, Getra; Gutman, Ivan.
In: Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences, Vol. 57, No. 3-4, 2002, p. 143-153.

Research output: Journal contributionsJournal articlesResearchpeer-review

Harvard

APA

Vancouver

Bibtex

@article{bbe7c6dde03a4f37b86b70e91803c8bc,
title = "On kites, comets, and stars. Sums of eigenvector coefficients in (molecular) graphs.",
abstract = "Two graph invariants were encountered that form the link between (molecular) walk counts and eigenvalues of graph adjacency matrices. In particular, the absolute value of the sum of coefficients of the first or principal (normalized) eigenvector, s 1, and the analogous quantity s n, pertaining to the last eigenvector, appear in equations describing some limits (for infinitely long walks) of relative frequencies of several walk counts. Quantity s 1 is interpreted as a measure of mixedness of a graph, and s n, which plays a role for bipartite graphs only, is interpreted as a measure of the imbalance of a bipartite graph. Consequently, s n is maximal for star graphs, while the minimal value of s n is zero. Mixedness s 1 is maximal for regular graphs. Minimal values of s 1 were found by exhaustive computer search within the sample of all simple connected undirected n-vertex graphs, n ≤ 10: They are encountered among graphs called kites. Within the special sample of tree graphs (searched for n ≤ 20) so-called double snakes have maximal s 1, while the trees with minimal s 1 are so-called comets. The behaviour of stars and double snakes can be described by exact equations, while approximate equations for s 1 of kites and comets could be derived that are fully compatible with and allow to predict some pecularities of the results of the computer search. Finally, the discriminating power of s 1, determined within trees and 4-trees (alkanes), was found to be high. ",
keywords = "Mathematics, Eigenvector Coefficients, Molecular Graphs, Walks",
author = "Christoph R{\"u}cker and Getra R{\"u}cker and Ivan Gutman",
year = "2002",
doi = "10.1515/zna-2002-3-406",
language = "English",
volume = "57",
pages = "143--153",
journal = "Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences",
issn = "0932-0784",
publisher = "Walter de Gruyter GmbH",
number = "3-4",

}

RIS

TY - JOUR

T1 - On kites, comets, and stars. Sums of eigenvector coefficients in (molecular) graphs.

AU - Rücker, Christoph

AU - Rücker, Getra

AU - Gutman, Ivan

PY - 2002

Y1 - 2002

N2 - Two graph invariants were encountered that form the link between (molecular) walk counts and eigenvalues of graph adjacency matrices. In particular, the absolute value of the sum of coefficients of the first or principal (normalized) eigenvector, s 1, and the analogous quantity s n, pertaining to the last eigenvector, appear in equations describing some limits (for infinitely long walks) of relative frequencies of several walk counts. Quantity s 1 is interpreted as a measure of mixedness of a graph, and s n, which plays a role for bipartite graphs only, is interpreted as a measure of the imbalance of a bipartite graph. Consequently, s n is maximal for star graphs, while the minimal value of s n is zero. Mixedness s 1 is maximal for regular graphs. Minimal values of s 1 were found by exhaustive computer search within the sample of all simple connected undirected n-vertex graphs, n ≤ 10: They are encountered among graphs called kites. Within the special sample of tree graphs (searched for n ≤ 20) so-called double snakes have maximal s 1, while the trees with minimal s 1 are so-called comets. The behaviour of stars and double snakes can be described by exact equations, while approximate equations for s 1 of kites and comets could be derived that are fully compatible with and allow to predict some pecularities of the results of the computer search. Finally, the discriminating power of s 1, determined within trees and 4-trees (alkanes), was found to be high.

AB - Two graph invariants were encountered that form the link between (molecular) walk counts and eigenvalues of graph adjacency matrices. In particular, the absolute value of the sum of coefficients of the first or principal (normalized) eigenvector, s 1, and the analogous quantity s n, pertaining to the last eigenvector, appear in equations describing some limits (for infinitely long walks) of relative frequencies of several walk counts. Quantity s 1 is interpreted as a measure of mixedness of a graph, and s n, which plays a role for bipartite graphs only, is interpreted as a measure of the imbalance of a bipartite graph. Consequently, s n is maximal for star graphs, while the minimal value of s n is zero. Mixedness s 1 is maximal for regular graphs. Minimal values of s 1 were found by exhaustive computer search within the sample of all simple connected undirected n-vertex graphs, n ≤ 10: They are encountered among graphs called kites. Within the special sample of tree graphs (searched for n ≤ 20) so-called double snakes have maximal s 1, while the trees with minimal s 1 are so-called comets. The behaviour of stars and double snakes can be described by exact equations, while approximate equations for s 1 of kites and comets could be derived that are fully compatible with and allow to predict some pecularities of the results of the computer search. Finally, the discriminating power of s 1, determined within trees and 4-trees (alkanes), was found to be high.

KW - Mathematics

KW - Eigenvector Coefficients

KW - Molecular Graphs

KW - Walks

UR - http://www.scopus.com/inward/record.url?scp=0036347081&partnerID=8YFLogxK

U2 - 10.1515/zna-2002-3-406

DO - 10.1515/zna-2002-3-406

M3 - Journal articles

VL - 57

SP - 143

EP - 153

JO - Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences

JF - Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences

SN - 0932-0784

IS - 3-4

ER -

Documents

DOI

Recently viewed

Researchers

  1. Moritz Meyer

Publications

  1. Final departure
  2. Better performance of organic than conventional tomato varieties in single and mixed cropping
  3. "Are we in sync with each other?" Exploring the effects of cosleeping on heterosexual couplestextquotesingle sleep using simultaneous polysomnography
  4. Sprache und Sprachgebrauch untersuchen in der Primarstufe
  5. Super-GAU und Computersimulation
  6. Das Simulationsexperiment
  7. Ungleichheit, Differenz und ‚Diversity’
  8. Resource selection by sympatric wild equids in the Mongolian Gobi
  9. Revidierbarkeit, ein Muster der Hypersphäre
  10. Risk aversion and labour market outcomes
  11. Non-linear effects of comparison income in quit decisions: status versus signal !
  12. The Bumpy Road from Investigation to Knowledge
  13. From railroad imperialism to neoliberal reprimarization: Lessons from regime-shifts in the Global Soybean Complex
  14. On the Question of Blockchain Activism
  15. Morden im Norden
  16. Changing learning environments at university?
  17. Nonlinear recurrence analysis of piezo sensor placement for unmanned aerial vehicle motor failure diagnosis
  18. Briefe schreiben in der Sekundarstufe I
  19. Alleviating depression in the unemployed
  20. Klassensprachen - Some Preliminary Theses
  21. Soziale Farbe (II)
  22. Links between media communication and local perceptions of climate change in an indigenous society
  23. Analysis of benzalkonium chloride in the effluent from European hospitals by solid-phase extraction and high-performance liquid chromatography with post-column ion-pairing and fluorescence detection
  24. Saproxylic beetles
  25. A novel radio-frequency plasma probe for monitoring systems in dielectric deposition processes
  26. Nachhaltigkeitsinformatik
  27. The effects of visual sustainability labels on consumer perception and behavior
  28. Phasing-out Coal in Germany – Not Only, But Also a Legal Challenge
  29. Occurrence of the antidiabetic drug Metformin and its ultimate transformation product Guanylurea in several compartments of the aquatic cycle
  30. The Importance of Being First
  31. Normalitätskonstruktion und Selbstbilder erwachsener Reitender mit einer Körper- oder Sinnesbehinderung
  32. Schätzen – Aber wie?
  33. The universe of group representation in Germany
  34. The need for pluralism in landscape models: a reply to Dunn and Majer
  35. Schreiben
  36. Engaging with entrepreneurial diversity
  37. The ten principles of green sample preparation