Kekule patterns and Clar patterns in bipartite plane graphs

Abstract

Let G be a finite bipartite plane graph. In a chemical context, a set of pairwise disjoint edges that cover all vertices of G (i.e., a perfect matching of G) is called a Kekulé pattern of G, and a set of pairwise disjoint cells of G such that the deletion of all vertices incident to these cells results in a graph that has a Kekulé pattern, or is empty, is called a Ciar pattern of G. Let k(G) and c(G) denote the number of Kekulé patterns and of Ciar patterns of G, respectively. It is shown that k(G) is not smaller than c(G) and that equality holds if G is an outerplane graph. This result generalizes a well-known proposition of the theory of benzenoid hydrocarbons; the proof uses a new idea.

Push-relabel based algorithms for the maximum transversal problem

“…The maximum cardinality bipartite matching problem has numerous applications in diverse domains, including scheduling [3], timetabling [4,5], image processing [6], and chemical structure analysis [7]. Burkard et al [8,Ch.…”

Advances of Clar's Aromatic Sextet Theory and Randic´'s Conjugated Circuit Model

“…Let ( ) r G denote the number of resonant patterns of G . John et al [60] obtained Theorem 2.8; and a refined result --Theorem 2.9 was obtained in [61,62].…”

“…Since it has the Fries structure so that each hexagon is alternating, any set of disjoint hexagons always forms a sextet pattern. Based upon this, Shiu et al [68] computed the sextet polynomial of 60 C as 8 Ye et al [69] showed that every hexagon of a fullerene is resonant, determined all the other eight 3-resonant fullerenes (i.e. every set of at most three disjoint hexagons forms a sextet pattern) and proved that any independent hexagons of a 3-resonant fullerene graph form a sextet pattern.…”

“…For benzenoid hydrocarbons, both the Clar number and Kekulé count can measure their stabilities. However, Austin et al [71] constructed 20 distinct fullerene isomers of 60 C whose Kekulé counts surpass the Kekulé count (12500) of icosahedral 60 C . So, the maximality of Kekulé counts of fullerene isomers may not correspond to the highest stability.…”

GPU Accelerated Maximum Cardinality Matching Algorithms for Bipartite Graphs

“…The problem asks for a maximum set of vertex disjoint edges in a given bipartite graph. It has many applications in a variety of fields such as image processing [18], chemical structure analysis [16], and bioinformatics [2] (see also another two discussed by Burkard et al [4,Section 3.8]). Our motivating application lies in solving sparse linear systems of equations, as algorithms for computing a maximum cardinality bipartite matching are run routinely in the related solvers.…”