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![The resonance graph R(G;F)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R(G;{\mathcal {F}})$$\end{document} of the coronene G](profile/Dong-Ye-4/publication/320179899/figure/fig3/AS:11431281180786266@1691718931850/The-resonance-graph-RGFdocumentclass12ptminimal-usepackageamsmath.png)
The resonance graph R(G;F)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R(G;{\mathcal {F}})$$\end{document} of the coronene G
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Let $G$ be a graph embedded in a surface and let $\mathcal F$ be a set of even faces of $G$ (faces bounded by a cycle of even length). The resonance graph of $G$ with respect to $\mathcal F$, denoted by $R(G;\mathcal F)$, is a graph such that its vertex set is the set of all perfect matchings of $G$ and two vertices $M_1$ and $M_2$ are adjacent to...
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A polynomial associated with G is defined as \(t(G,w)=\sum _{T\in {\mathbb {T}}(G)}\prod _{e\in E(T)}w_e(G)\) (\({\mathbb {T}}(G)\) is the set of spanning trees of G), which is a weighted enumeration of spanning trees of graphs. It is known that any graph G is an intersection graph of a linear hypergraph, which corresponds to a clique partition of...
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Citations
... Then, in Section 6, the equality between the first ZZ polynomial of a phenylene and the generalized cube polynomial of its resonance graph is proved. Note that resonance graphs are used to model interactions among different Kekulé structures [30] and that the relation between the (generalized) Zhang-Zhang polynomial of a molecular graph and the (generalized) cube polynomial of its resonance graph was investigated in the past for other families of graphs [1,25,26,32,36]. For some recent research on resonance graphs see [3,5,6,8,27]. ...
... Remark. Note that Corollary 2 also follows by Theorem 1.5 from [25], where this equality was proved in a more general way. ...
The aim of this paper is to study some variations of the Zhang-Zhang polynomial for phenylenes, which can be obtained as special cases of the multivariable Zhang-Zhang polynomial. Firstly, we prove the equality between the first Zhang-Zhang polynomial of a phenylene and the generalized Zhang-Zhang polynomial of some benzenoid graph, which enables us to prove also the equality between the first Zhang-Zhang polynomial and the generalized cube polynomial of the resonance graph. Next, some results on the roots of the second Zhang-Zhang polynomial of phenylenes are provided and another expression for this polynomial is established. Finally, we give structural interpretation for (partial) derivatives of different Zhang-Zhang polynomials.
... It is interesting that the Clar number, the number of Kekulé structures, and the first Herndon number can be easily calculated by using this polynomial. For some recent papers on the Zhang-Zhang polynomial, see [10][11][12]. ...
... These graphs are used to model interactions among different Kekulé structures; see [13] for a survey on this topic. It is worth mentioning that the relation between the Zhang-Zhang polynomial of a molecular graph and the cube polynomial of its resonance graph was considered for different families of graphs [12,14]. Some recent research on resonance graphs can be found, for example, in [15]. ...
The Zhang–Zhang polynomial of a benzenoid system is a well-known counting polynomial that was introduced in 1996. It was designed to enumerate Clar covers, which are spanning subgraphs with only hexagons and edges as connected components. In 2018, the generalized Zhang–Zhang polynomial of two variables was defined such that it also takes into account 10-cycles of a benzenoid system. The aim of this paper is to introduce and study a new variation of the Zhang–Zhang polynomial for phenylenes, which are important molecular graphs composed of 6-membered and 4-membered rings. In our case, Clar covers can contain 4-cycles, 6-cycles, 8-cycles, and edges. Since this new polynomial has three variables, we call it the multivariable Zhang–Zhang (MZZ) polynomial. In the main part of the paper, some recursive formulas for calculating the MZZ polynomial from subgraphs of a given phenylene are developed and an algorithm for phenylene chains is deduced. Interestingly, computing the MZZ polynomial of a phenylene chain requires some techniques that are different to those used to calculate the (generalized) Zhang–Zhang polynomial of benzenoid chains. Finally, we prove a result that enables us to find the MZZ polynomial of a phenylene with branched hexagons.
... counts the number of induced i -cubes in G . This polynomial was introduced in [3], see also [2,18]. ...
... This polynomial was introduced in [3], see also [2,18]. Clearly, c 0 (G) = |V (G)| and c 1 (G) = |E(G)|. ...
In this paper, $k$-Pell graphs $\Pi _{n,k}$, $k\ge 2$, are introduced such that $\Pi _{n,2}$ are the Pell graphs defined earlier by Munarini. Several metric, enumerative, and structural properties of these graphs are established. The generating function of the number of edges of $\Pi _{n,k}$ and the generating function of its cube polynomial are determined. The center of $\Pi _{n,k}$ is explicitly described; if $k$ is even, then it induces the Fibonacci cube $\Gamma_{n}$. It is also shown that $\Pi _{n,k}$ is a median graph, and that $\Pi _{n,k}$ embeds into some Fibonacci cube.
... A graph is bipartite if its vertex set can be partitioned into two subsets X and Y so that every edge has one end vertex in X and the other in Y . Tratnik and Ye [24] obtained the following result. ...
... Theorem 2.5. [24] Let G be a graph embedded in a surface and F ⊂ F (G). Then R(G; F ) is bipartite. ...
In a region $R$ consisting of unit squares, a domino is the union of two adjacent squares and a (domino) tiling is a collection of dominoes with disjoint interior whose union is the region. The flip graph $\mathcal{T}(R)$ is defined on the set of all tilings of $R$ such that two tilings are adjacent if we change one to another by a flip (a $90^{\circ}$ rotation of a pair of side-by-side dominoes). It is well-known that $\mathcal{T}(R)$ is connected when $R$ is simply connected. By using graph theoretical approach, we show that the flip graph of $2m\times(2n+1)$ quadriculated cylinder is still connected, but the flip graph of $2m\times(2n+1)$ quadriculated torus is disconnected and consists of exactly two isomorphic components. For a tiling $t$, we associate an integer $f(t)$, forcing number, as the minimum number of dominoes in $t$ that is contained in no other tilings. As an application, we obtain that the forcing numbers of all tilings in $2m\times (2n+1)$ quadriculated cylinder and torus form respectively an integer interval whose maximum value is $(n+1)m$.
