Distribution of K, and Kekulé Structure Statistics

Abstract

The problem to find all benzenoid systems which have a given number of Kekulé structures (K) for h ≤ 8 was posed relatively early (Gutman 1982c). On the other hand it was known at that time that infinitely many benzenoids have K=9. Until quite recently the only known theorem about this question was: If K{B} = 2, then B = benzene (Gutman 1983). In other words, there exists one benzenoid with K=2. Furthermore, it was conjectured that there is one benzenoid with K=3 (viz. naphthalene),one with K=4 (anthhacene),and that there are two with K=5 (napltithacene and phenanthrene). The distinction between normal and essentially disconnected benzenoids (Paragraph 2.3.2) was a break-through for this problem. Based on Theorem 13 (see Section 3.4), which was put forward recently by Gutman and Cyvin (1988), it was deduced that there exists a limited number of normal benzenoids with any K number (K > 1). The infinite number for K=9 (and other values of K > 9) is due to the essentially disconnected systems. Hence it has sense to ask, for instance: — How many normal benzenoids have K = 100 ? (The answer is 444.)