Closed-form formulas for the Zhang-Zhang polynomials of benzenoid structures: Prolate rectangles and their generalizations

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• 作者：Chien-Pin Chou^a ; ^{cpchou.imo95g@nctu.edu.tw" class="auth_mail" title="E-mail the corresponding author} ; Jin-Su Kang^b ; Henryk A. Witek^a
• 关键词：Perfect matching ; Clar cover ; Clar structure ; Zhang&ndash ; Zhang polynomial
• 刊名：Discrete Applied Mathematics
• 出版年：2016
• 出版时间：10 January 2016
• 年：2016
• 卷：198
• 期：Complete
• 页码：101-108
• 全文大小：4253 K

文摘

We show that the Zhang–Zhang (ZZ) polynomial of a benzenoid obtained by fusing a parallelogram k to view the MathML source">M(m,n)$M(m,n)$ with an arbitrary benzenoid structure k to view the MathML source">ABC$ABC$ can be simply computed as a product of the ZZ polynomials of both fragments. It seems possible to extend this important result also to cases where both fused structures are arbitrary Kekuléan benzenoids. Formal proofs of explicit forms of the ZZ polynomials for prolate rectangles k to view the MathML source">Pr(m,n)$Pr(m,n)$ and generalized prolate rectangles k to view the MathML source">Pr([m₁,m₂,…,m_n],n)$Pr([m_1,m_2,\dots ,m_n],n)$ follow as a straightforward application of the general theory, giving k to view the MathML source">ZZ(Pr(m,n),x)=(1+(1+x)⋅m)ⁿ$ZZ(Pr(m,n),x)={(1+(1+x)\cdot m)}^n$ and $ZZ(Pr([m_1,m_2,\dots ,m_n],n),x)={\prod }_{\mathrm{<font\; color="red">k</font>}=1}^n(1+(1+x)\cdot m_{\mathrm{<font\; color="red">k</font>}})$.