The Gutman Formulas for Algebraic Structure Count

Abstract

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” Kekulé structures of a conjugated molecule. Precisely, algebraic structure count (ASC-value) of the bipartite graph G corresponding to the skeleton of a conjugated hydrocarbon is defined by $ASC\{ G\} = \sqrt {|\det A|}$ where A is the adjacency matrix of G. In the case of bipartite planar graphs containing only circuits of the length of the form 4s+2 (s=1,2,...) (the case of benzenoid hydrocarbons), this number is equal to the number of the perfect matchings (K-value) of G. However, if some of circuits are of the length 4s (s=1,2,...) then the problem of evaluation ASC-value becomes more complicated. The theorem formulated and proved in this paper gives a simple and efficient algorithm for calculation of algebraic structure count of an arbitrary bipartite graphs with n+n vertices. Three recurrence formulas for the algebraic structure count – the Gutman formulas, which are closely analogous to the well-known recurrence formula K{G}=K{G−e}+K{G−(e)} for the number of perfect matchings (G−e is the subgraph obtained from the graph G by deleting the edge e and G−(e) is the subgraph obtained from G by deleting both the edge e and its terminal vertices) are obtained as a simple corollary of the theorem.