Graph Theory With Applications
Graph Theory With Applications
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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) 485-490; 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, ]. 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, ; 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.