DOI: 10.1007/s10910-006-9074-0 Journal of Mathematical Chemistry, Vol. 41, No. 3, April 2007 (© 2006) Algebraic structure count of some cyclic hexagonal-square chains on the M¨ obius strip Olga Bodroˇ za-Panti´ c Department of Mathematics and Informatics, Trg D. Obradovi ´ ca 4, University of Novi Sad, 21000 Novi Sad, Serbia E-mail: bodroza@im.ns.ac.yu Received 1 November 2005; revised 10 December 2005 The concept of ASC (Algebraic structure count) is introduced into theoretical organic chemistry by Wilcox as the difference between the number of so-called “even” and “odd” Kekul´ e structures of a conjugated molecule. Precisely, algebraic structure count (ASC-value) of the bipartite graph G corresponding to the skeleton of a con- jugated hydrocarbon is defined by ASC{G} def =  | det A | where A is the adjacency matrix of G. The determination of algebraic structure count of (bipartite) cyclic hex- agonal-square chains in the the class of plane such graphs is known. In this paper we expand these considerations on the non-plane class. An explicit combinatorial formula for ASC is deduced in the special case when all hexagonal fragments are isomorphic. KEY WORDS: algebraic structure count, Kekul´ e structure AMS subject classification (2001): 05C70, 05C50, 05B50, 05A15 1. Introduction The algebraic structure count (ASC-value) of a bipartite graph G is defined by ASC{G} def =  | det A |, where A is the adjacency matrix of G. The thermodynamic stability of an alternant hydrocarbon is related to the ASC-value for the bipartite graph which represents its skeleton. The basic appli- cation of ASC is in the following. Among two isomeric conjugated hydrocarbons (whose related graphs have an equal number of vertices and an equal number of edges), the one having greater ASC will be more stable. In particular, if ASC = 0, then the respective hydrocarbon is extremely reactive and usually does not exist [4, 5]. In the case of the bipartite plane graphs containing only circuits of the length of the form 4s + 2 (s = 1, 2, . . .) (benzenoid hydrocarbons) all perfect 283 0259-9791/07/0400-0283/0 © 2006 Springer Science+Business Media, Inc.