Resonance in catacondensed benzenoid hydrocarbons

Randić, Milan

doi:10.1002/(sici)1097-461x(1997)63:2<585::aid-qua28>3.0.co;2-5

Cited by 36 publications

(20 citation statements)

References 31 publications

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“…3. The concept of the resonance graph appears to be very natural in chemistry, see [17,18] and the arguments from the introduction of [10]. The structure of the resonance graphs of hexagonal graphs was closely examined in [11] where the so called decomposition theorem is proved.…”

Section: Theorem 1 Let G Be An Arbitrary Fibonaccene With H Hexagonsmentioning

confidence: 99%

Fast Recognition of Fibonacci Cubes

Taranenko

Vesel

2007

Algorithmica

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Section: Theorem 1 Let G Be An Arbitrary Fibonaccene With H Hexagonsmentioning

confidence: 99%

Fast Recognition of Fibonacci Cubes

Taranenko

Vesel

2007

Algorithmica

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“…The Z-transformation graph (also called resonance graph) is introduced independently by Gründler [10], Zhang et al [37], Randić [26] and Fournier [7], and widely studied by Zhang and Zhang [44,45], Kalvažar et al [15,16,17], Lam and Zhang [20], Zhang et al [38,40,42] andŽigert Pleteršek [47,48]. Recently, Zhang et al [41] introduced the concept of matchable distributive lattice and got some consequences on matchable distributive lattices, Yao and Zhang [34] obtained some results on non-matchable distributive lattices with a cut-element.…”

Section: Introductionmentioning

confidence: 99%

Structure and enumeration results of matchable Lucas cubes

Wang

Zhao

Yao

2020

Discrete Applied Mathematics

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A lucasene is a hexagon chain that is similar to a fibonaccene, an L-fence is a poset the Hasse diagram of which is isomorphic to the directed inner dual graph of the corresponding lucasene. A new class of cubes, which named after matchable Lucas cubes according to the number of its vertices (or elements), are a series of directed or undirected Hasse diagrams of filter lattices of L-fences. The basic properties and several classes of polynomials, e.g. rank generating functions, cube polynomials and degree sequence polynomials, of matchable Lucas cubes are obtained. Some special conclusions on binomial coefficients and Lucas triangle are given.A set P equipped with a binary relation ≤ satisfying reflexivity, antisymmetry and transitivity is said to be a partially ordered set (poset for short). Given any poset P , the dual P * of P is formed by defining x ≤ y to hold in P * if and only if y ≤ x holds in P . A subposet S of P is a chain if any two elements of S are comparable, and denoted by n if |S| = n [3]. Let x ≺ y denote y cover s x in P , i.e. x < y and. The set of all filters of a poset P is denoted by F (P ), and carries the usual anti-inclusion order, forms a finite distributive lattice [3,31] called filter lattice. For a finite lattice L, we denote by0 L (resp.1 L ) the minimum (resp. maximum) element in L.The symmetric difference of two finite sets A and B is defined asis a perfect matching of a graph and C is an M -alternating cycle of the graph, then the symmetric difference of M and edge-set E(C) is another perfect matching of the graph, which is simply denoted by M ⊕ C. Let G be a plane bipartite graph with a perfect matching, and the vertices of G are colored properly black and white such that the two ends of every edge receive different colors. An M -alternating cycle of G is said to be proper, if every edge of the cycle belonging to M goes from white end-vertex to black end-vertex by the clockwise orientation of the cycle; otherwise improper [43]. An inner face of a graph is called a cell if itsboundary is a cycle, and we will say that the cycle is a cell too. For some concepts and notations not explained in the paper, refer to [3,8,31] for poset and lattice, [1,11] for graph theory. Zhang and Zhang [45] extended the concept of Z-transformation graph of a hexagonal system to plane bipartite graphs. Definition 2.1 ([45]) Let G be a plane bipartite graph. The Z-transformation graph Z(G) is defined on M(G): M 1 , M 2 ∈ M(G) are joined by an edge if and only if M 1 ⊕ M 2 is a cell of G. And Z-transformation digraph Z(G) is the orientation of Z(G): an edge M 1 M 2 of Z(G) is oriented from M 1 to M 2 if M 1 ⊕ M 2 form a proper M 1 -alternating (thus improper M 2 -alternating) cell. Let G be a bipartite graph, from Theorem 4.1.1 in [21], we have that G is elementary if and only if G is connected and every edge of G lies in a perfect matching of G. Let G be a plane bipartite graph with a perfect matching, a binary relation ≤ on M(G) is defined as: for M 1 , M 2 ∈ M(G), M 1 ≤ M 2 if and only if Z(G) has a d...

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“…Since 1980 ′ s there have been developed a combinatorial object, the Z-transformation graph (or resonance graph) [23,24] on the set perfect matchings of a hexagonal system, late extended to a general plane bipartite graph [27,33,34,35]; see a recent survey [29]. Randić [15,16] showed that the leading eigenvalue of the resonance graphs has a quite satisfactory correlation with the resonance energy of benzenoid hydrocarbons.…”

Section: Introductionmentioning

confidence: 99%

Decomposition theorem on matchable distributive lattices

Zhang

Yang

Yao

2014

Discrete Applied Mathematics

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A distributive lattice structure ${\mathbf M}(G)$ has been established on the set of perfect matchings of a plane bipartite graph $G$. We call a lattice {\em matchable distributive lattice} (simply MDL) if it is isomorphic to such a distributive lattice. It is natural to ask which lattices are MDLs. We show that if a plane bipartite graph $G$ is elementary, then ${\mathbf M}(G)$ is irreducible. Based on this result, a decomposition theorem on MDLs is obtained: a finite distributive lattice $\mathbf{L}$ is an MDL if and only if each factor in any cartesian product decomposition of $\mathbf{L}$ is an MDL. Two types of MDLs are presented: $J(\mathbf{m}\times \mathbf{n})$ and $J(\mathbf{T})$, where $\mathbf{m}\times \mathbf{n}$ denotes the cartesian product between $m$-element chain and $n$-element chain, and $\mathbf{T}$ is a poset implied by any orientation of a tree.Comment: 19 pages, 7 figure

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Resonance in catacondensed benzenoid hydrocarbons

Cited by 36 publications

References 31 publications

Fast Recognition of Fibonacci Cubes

Fast Recognition of Fibonacci Cubes

Structure and enumeration results of matchable Lucas cubes

Decomposition theorem on matchable distributive lattices

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