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莫比乌斯梯子


MoebiusLadders

莫比乌斯梯子,有时也称为莫比乌斯轮(Jakobson 和 Rivin 1999),阶数为 n 的莫比乌斯梯子是通过在阶数为 n 的棱柱图 中引入扭曲而获得的简单图,该棱柱图同构于循环图 Ci_(2n)(1,n)。莫比乌斯梯子有时表示为 M_n

4-莫比乌斯梯子被称为瓦格纳图(2n-1)-莫比乌斯梯子梯级图同构于 Haar 图 H(2^(2n)+1)

莫比乌斯梯子是哈密顿图优美图 (Gallian 1987, Gallian 2018),并且根据构造,是单交叉图。莫比乌斯梯子也是非平凡的双平面图

对于 n=3, 4, ...,有向哈密顿环的数量为 12, 10, 16, 14, 20, 18, 24, ... (OEIS A124356),由闭合形式给出

 |HC(n)|=2[(n+2)-(-1)^n].
(1)

n-莫比乌斯梯子图具有独立多项式

 I_n(x)=2^(-n)[-2^n(-x)^n+(x-sqrt(x(x+6)+1)+1)^n+(x+sqrt(x(x+6)+1)+1)^n].
(2)

独立多项式匹配多项式的递推方程由下式给出

I_n(x)=I_(n-1)(x)+x(x+2)I_(n-2)(x)+x^2I_(n-3)(x)
(3)
mu_n(x)=(x^2-1)mu_(n-1)(x)+2(1-x^2)mu_(n-2)(x)+(x^2+1)mu_(n-3)(x)-mu_(n-4)(x).
(4)

n-莫比乌斯梯子的二部双图棱柱图 Y_(2n)M_n图平方循环图 Ci_(2n)(1,2,n-1,n),其图立方Ci_(2n)(1,2,3,n-2,n-1,n)


另请参阅

循环图, 交叉棱柱图, Helm 图, 梯子图, 棱柱图, 瓦格纳图, Web 图, 轮图

使用 Wolfram|Alpha 探索

参考文献

Biggs, N. L. Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, pp. 20-21, 1993.Gallian, J. "Labeling Prisms and Prism Related Graphs." Congr. Numer. 59, 89-100, 1987.Gallian, J. "Dynamic Survey of Graph Labeling." Elec. J. Combin. DS6. Dec. 21, 2018. https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6.Godsil, C. and Royle, G. Algebraic Graph Theory. New York: Springer-Verlag, pp. 118 and 131, 2001.Hladnik, M.; Marušič, D.; and Pisanski, T. "Cyclic Haar Graphs." Disc. Math. 244, 137-153, 2002.McSorley, J. P. "Counting Structures in the Moebius Ladder." Disc. Math. 184, 137-164, 1998.Jakobson, D. and Rivin, I. "On Some Extremal Problems in Graph Theory." 8 Jul 1999. http://arxiv.org/abs/math.CO/9907050.Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, pp. 263 and 270, 1998.Sloane, N. J. A. Sequence A124356 in "The On-Line Encyclopedia of Integer Sequences."

请引用为

Weisstein, Eric W. “莫比乌斯梯子。” 来自 MathWorld--Wolfram Web 资源。 https://mathworld.net.cn/MoebiusLadder.html

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