An upper bound on the independence number of benzenoid systems

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, 〈〈http://www.math.uh.edu/∼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/∼siemion/pony.html〉〉; 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.