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基尔霍夫指数

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基尔霍夫指数 Kf, 也被简称为电阻，并用 R 表示 (Lukovits et al. 1999), 一个 连通图 G 在 n 个节点上的定义为

Kf=1/2sum_(i=1)^nsum_(j=1)^n(Omega)_(ij),

其中 (Omega)_(ij) 是 电阻距离 矩阵。

除非另有说明，氢原子通常在计算有机化学家在将苯环写成六边形时通常会忽略的此类指标时被忽略 (Devillers and Balaban 1999, p. 25)。

许多图的预计算值在 Wolfram 语言 中实现为GraphData[g,"KirchhoffIndex"].

下表总结了各种特殊图类的基尔霍夫指数值。

图类OEISKf(G_1), Kf(G_2), ...
Andrásfai 图A000000/A0000001, 10, 134/7, 3080/109, 263599/7059, 5244806/113017, ...
反棱柱图A000000/A000000X, X, 13/2, 290/21, 551/22, 41, ...
阿波罗尼安网络A000000/A0000003, 834/85, 30154/475, 23555722/44125, 1259601793/263125, ...
鸡尾酒会图 K_(n×2)A000000/A000000infty, 5, 13/2, 25/3, 41/4, 61/5, 85/6, 113/7, 145/8, 181/9, ...
完全二分图 K_(n,n)A0000001, 5, 9, 13, 17, 21, 25, 29, 33, 37, ...
完全图 K_nA0014770, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
完全三部图 K_(n,n,n)A0000002, 13/2, 11, 31/2, 20, 49/2, ...
交叉棱柱图A000000X, 58/3, 53, 332/3, 595/3, 322, 1463/3, 2104/3, 969, 3890/3, ...
皇冠图A000000/A000000X, X, 35/2, 58/3, 271/12, 131/5, 899/30, 710/21, ...
立方体连接环图A000000/A000000X, X, 346, 6442544/2415, 33760334655251468654052195628373/1958536428315956487415847430, ...
环图 C_nA138190/A138191X, X, 2, 5, 10, 35/2, 28, 42, ...
折叠立方体图A000000/A000000X, 1, 3, 13, 50, 548/3, 1960/3, 6968/3, 8272, ...
齿轮图A000000/A000000X, X, 18, 69/2, 1085/19, 257/3, 8526/71, 6733/42, 10935/53, 53945/209, ...
网格图 P_n square P_nA000000/A0000000, 5, 69/2, 884/7, 11155/33, 2520, 5488, 10752, ...
网格图 P_n square P_n square P_nA000000/A0000000, 58/3, 16959/70, 489296/357, 119593165825/23110593, 136080, ...
半立方体图A290365/A2903660, 1, 3, 25/3, 70/3, 3014/45, 2968/15, 63148/105, 197072/105, ...
超立方体图 Q_nA290343/A2903441, 5, 58/3, 206/3, 3548/15, ...
莫比乌斯梯子 M_nA000000/A000000X, X, 9, 134/7, 1135/33, 725/13, 10367/123, 11732/97, 2835/17, ...
Mycielski 图A000000/A0000000, 1, 10, 4545/139, 8808777389/93842615, ...
奇图 O_nA000000/A0000000, 2, 33, 373, 4000, 264001/6, ...
平底锅图A000000/A000000X, X, 19/3, 23/2, 19, 88/3, 43, ...
路径图 P_nA0000000, 1, 4, 10, 20, 35, 56, 84, 120, 165, ...
排列星图 PS_nA000000/A0000000, 1, 35/2, 1314/5, 26520, 963598/7, ...
棱柱图 Y_nA000000/A000000X, X, 47/5, 58/3, 655/19, 279/5, ...
车图 K_n square K_nA000000/A000000X, 5, 18, 42, 80, 135, 210, 308, 432, ...
星图 S_nA0002900, 1, 4, 9, 16, 25, 36, 49, 64, 81, ...
太阳图A000000/A000000X, X, 65/6, 302/15, 75, 114, 161, 216, 279, 350, 429, ...
日瓣图 C_n circledot K_1A000000/A000000X, X, 23, 48, 105, 174, 259, 376, 513, 690, 891, ...
四面体图A000000/A000000X, X, X, X, X, 43, 207/2, 1945/9, 4901/12, 2149/3, ...
三角形图A000000/A000000X, 0, 2, 13/2, 57/4, 26, 85/2, 129/2, 371/4, 128, ...
网状图A000000/A000000X, X, 222/5, 173/2, 2780/19, 4521/20, 23282/71, 3179/7, 160848/265, ...
轮图 W_nA000000/A000000X, X, X, 16/3, 95/11, 129/10, ...

封闭形式总结在下表中。 环图 由 Klein et al. (1995) 和 Babić et al. (2002) 考虑。这里，H_n 是 调和数，而 Phi(z,s,a) 是 Lerch 超越函数。

图Kf(G_n)
鸡尾酒会图 K_(n×2)(2n^2-2n+1)/(n-1)
完全二分图 K_(n,n)4n-3
完全图 K_nn-1
完全三部图 K_(n,n,n)1/2(9n-5)
交叉棱柱图1/3(3n^3-10n+7)
皇冠图(4n^3-11n^2+10n-4)/((n-2)(n-1))
环图 C_n，其中 n>=31/(12)(n-1)n(n+1)
空图 K^__n{0 for n=1; infty otherwise
超立方体图 Q_n-2^(n-1)H_n-4^nR[Phi(z,1,n+1)]
平底锅图1/(12)(n^3+2n^2+11n-2)
路径图 P_n1/6(n-1)n(n+1)
车图 K_n square K_n1/2(n-1)n(n+3)
星图 S_n(n-1)^2
三角形图1/8(n-2)(n^2+3n-2)

另请参阅

Balaban 指数, 图距离矩阵, 基尔霍夫和指数, 电阻距离, Wiener 指数, Wiener 和指数

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参考文献

Babić, D.; Klein, D. J.; Lukovits, I.; Nikolić, S.; and Trinajstić, N. "Resistance-Distance Matrix: A Computational Algorithm and Its Applications." Int. J. Quant. Chem. 90, 166-176, 2002.Bonchev, D.; Balaban, A. T.; Liu, X.; and Klein, D. J. "Molecular Cyclicity and Centricity of Polycyclic Graphs. I. Cyclicity Based on Resistance Distances or Reciprocal Distances." Int. J. Quan. Chem. 50, 1-20, 1994.Devillers, J. and Balaban, A. T. (Eds.). Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, pp. 40-41 and 114, 1999.Klein, D. J.; Lukovits, I.; and Gutman, I. "On the Definition of the Hyper-Wiener Index for Cycle-Containing Structures." J. Chem. Inf. Comput. 35, 50-52, 1995.Lukovits, I.; Nikolić, S.; and Trinajstić, N. "Resistance Distance in Regular Graphs." Int. J. Quan. Chem. 71, 217-225, 1999.Palacios, J. L. "Closed-Form Formulas for Kirchhoff Index." Int. J. Quant. Chem. 81, 135-140, 2001.Sloane, N. J. A. Sequences A000290/M3356, A001477, A138190, A138191, A290343, A290344, A290347, A290348, A290365, and A290366 in "The On-Line Encyclopedia of Integer Sequences."

在 Wolfram|Alpha 中被引用

基尔霍夫指数

请引用为

Weisstein, Eric W. "Kirchhoff Index." 来自 MathWorld--Wolfram 网络资源。 https://mathworld.net.cn/KirchhoffIndex.html

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