Anacondensed hexagonal systems

Abstract

Let HSh be the set of hexagonal systems with h hexagons and let ni(H) denote the number of internal vertices of H∈HSh. It is well known that for every H∈HSh:(1)0≤ni(H)≤2h+1−⌈12h−3⌉.The hexagonal systems which attain the lower bound of (1) are called catacondensed hexagonal systems, a class which has been extensively studied. On the other extreme we have the hexagonal systems which attain the upper bound of (1), which in contrast to the catacondensed hexagonal systems, we call anacondensed hexagonal systems, and denote them by Ah. We shall see in this paper that the number of anacondensed hexagonal systems in HSh have a super-polynomial growth for infinite values of h: for any t>0, there exist h large enough such that there are more than ht anacondensed hexagonal systems in HSh. Consequently, Ah is a large class in HSh.One useful parameter associated to a hexagonal system is the so-called number of bay regions of H∈HSh, denoted by b(H). We will show how to construct anacondensed hexagonal systems with a given number of bay regions. Moreover, among all hexagonal systems in Ah, we find those which have extremal value of number of bay regions. This result has strong implications in the study of the extremal values of vertex-degree-based topological indices (molecular descriptors) over Ah.