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An upper bound on the independence number of benzenoid systems

机译：苯类系统独立性数的上限

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Recently, the graph theoretic independence number has been linked to fullerene stability [S. Fajtlowicz, C. Larson, Graph-theoretic independence as a predictor of fullerene stability, Chem. Phys. Lett. 377 (2003) 4:35-490; S. Fajtlowicz, Fullerene Expanders, A list of Conjectures of Minuteman, Available from S. Fajtlowicz: math0@bayou.uh.edu]. In particular, stable fullerenes seem to minimize their independence numbers. A large piece of evidence for this hypothesis comes from the fact that stable benzenoids-close relatives of fullerenes-do minimize their independence numbers [S. Fajtlowicz, "Pony Express"-Graffiti's conjectures about carcinogenic and stable benzenoids, < http://www.math.uh.edu/similar to siemion/pony.html >]. In this paper, an upper bound on the independence number of benzenoids is introduced and proven-giving a limit on how large the independence ratio for benzenoids can be. In conclusion, this bound on independence is correlated to an upper bound on the number of unpaired sites a benzenoid system has with respect to a maximum matching, which is precisely the number of zero eigenvalues in the spectrum of the adjacency matrix (due to a conjecture of Graffiti and its proof by Sachs [S. Fajtlowicz, "Pony Express"-Graffiti's conjectures about carcinogenic and stable benzenoids, < http://www.math.uh.edu/similar to siemion/pony.htmll >; H. Sachs, P. John, S. Fajtlowicz, On Maximum Matchings and Eigenvalues of Benzenoid Graphs, preprint-MATCH]). Thus, since zero eigenvalues and unpaired sites are indicative of instability (reactivity), we get a simple but intuitive bound on how reactive a benzenoid molecule can be. (C) 2007 Published by Elsevier B.V.
机译：最近，图论的独立性数已经与富勒烯的稳定性相关联[S.S. Fajtlowicz，C. Larson，图论独立性作为富勒烯稳定性的预测因子，化学。物理来吧377（2003）4：35-490; S. Fajtlowicz，富勒烯膨胀剂，民兵猜想清单，可从S. Fajtlowicz获得：math0@bayou.uh.edu]。特别地，稳定的富勒烯似乎使它们的独立性数最小化。这一假设的大量证据来自这样一个事实，即稳定的苯环类化合​​物（富勒烯的近亲）确实使它们的独立性数最小化[S. Fajtlowicz，“小马快车” -Graffiti关于致癌和稳定的本泽酮的猜想，]。在本文中，介绍了苯并类化合物的独立性数的上限，并证明了对苯并类化合物的独立性可以达到多大的限制。总而言之，这个独立性的界限与本性系统相对于最大匹配的不成对位点数目的上限相关，这恰好是邻接矩阵频谱中零本征值的数目（由于猜想）涂鸦及其萨克斯的证明[S. Fajtlowicz，“小马快车”-涂鸦关于致癌和稳定的本烯类的猜想，; H. Sachs ，P。John，S。Fajtlowicz，《关于Benzenoid图的最大匹配和特征值》，preprint-MATCH]）。因此，由于零特征值和不成对的位点指示不稳定性（反应性），因此我们获得了一个简单而直观的界限，即对苯甲酸酯分子的反应能力。 （C）2007由Elsevier B.V.发布

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