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电阻距离

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图 G 中顶点 i 和 j 之间的电阻距离定义为当每个图边都被单位电阻器替换时，两个顶点之间的有效电阻（如同在它们之间连接电池时）（Klein 和 Randić 1993，Klein 2002）。这种电阻距离是图上的一个度量（Klein 2002）。

令 Omega_(ij) 为具有 n 个节点的连通图 G 中顶点 i 和 j 之间的电阻距离，并定义

Gamma=L+(1)/n,
(1)

其中 L 是 G 的拉普拉斯矩阵，1 是单位 n×n 矩阵。那么电阻距离矩阵由下式给出

(Omega)_(ij)=(Gamma)_(ii)^(-1)+(Gamma)_(jj)^(-1)-2(Gamma)_(ij)^(-1),
(2)

其中 A^(-1) 表示矩阵逆 (Babić 等人 2002)。这可以显式地写成

Omega=[2Gamma_(11)^(-1) Gamma_(11)^(-1)+Gamma_(22)^(-1) ... Gamma_(11)^(-1)+Gamma_(nn)^(-1); Gamma_(22)^(-1)+Gamma_(11)^(-1) 2Gamma_(22)^(-1) ... Gamma_(22)^(-1)+Gamma_(nn)^(-1); | | ... |; Gamma_(nn)^(-1)+Gamma_(11)^(-1) Gamma_(nn)^(-1)+Gamma_(22)^(-1) ... 2Gamma_(nn)^(-1)].
(3)

具有相同电阻距离集的图被称为电阻等价图。最小的此类图对具有九个顶点。

例如，四面体图的电阻距离矩阵为

Omega(K_4)=[0 1/2 1/2 1/2; 1/2 0 1/2 1/2; 1/2 1/2 0 1/2; 1/2 1/2 1/2 0]
(4)

而立方体图的电阻距离矩阵由下式给出

Omega(Q_3)=[0 7/(12) 7/(12) 3/4 7/(12) 3/4 3/4 5/6; 7/(12) 0 3/4 7/(12) 3/4 7/(12) 5/6 3/4; 7/(12) 3/4 0 7/(12) 3/4 5/6 7/(12) 3/4; 3/4 7/(12) 7/(12) 0 5/6 3/4 3/4 7/(12); 7/(12) 3/4 3/4 5/6 0 7/(12) 7/(12) 3/4; 3/4 7/(12) 5/6 3/4 7/(12) 0 3/4 7/(12); 3/4 5/6 7/(12) 3/4 7/(12) 3/4 0 7/(12); 5/6 3/4 3/4 7/(12) 3/4 7/(12) 7/(12) 0].
(5)
ResistanceMatrixPlatonic

柏拉图多面体图的电阻距离 (Klein 2002) 总结在下表中，以公分母表示，并在上面以图形方式说明。 Jeans (1925) 考虑了十二面体图的情况。

立体分母排序后的电阻距离
立方体图127, 9, 10
十二面体图3019, 27, 32, 34, 35
二十面体图3011, 14, 15
八面体图125, 6
四面体图21
ResistanceMatrixArchimedean

类似地，阿基米德立体的电阻距离在下面给出，并在上面以图形方式说明。

立体分母排序后的电阻距离
立方八面体图2411, 14, 15, 16
大斜方二十面截半十二面体图267514380166172084, 173751140, 190646963, 221685105, 272372574, 295109742, 301338668, 320673518, 345148397, 354812283, 361971116, 369550172, 381064593, 390079665, 394156361, 403801761, 405280440, 413491211, 417927248, 423905327, 430313930, 431484383, 431615693, 435250932, 438762291, 442133634, 445951845, 447430524, 456438590, 457489082, 458175207, 462416669, 463372068, 470296886, 476034686, 476835425, 478444515, 478664382, 483745052, 485853936, 486805896, 493083218, 497108172, 497550579, 499061297, 502747440, 503089004, 505386815, 506941514, 509320803, 511182242, 513097181, 514936860, 515575173, 516510357, 517043121, 520894371, 521353535, 521707218, 522228251, 523782950, 525033803, 528672702, 529607886, 530101733, 531714147, 533238108, 535548332, 537089358, 538353884, 540120215, 540275613, 540864390, 541799574, 542466050
大斜方截半立方八面体图10296063859, 65767, 72004, 84288, 102999, 108723, 113755, 118948, 127093, 129019, 130927, 130977, 136755, 137289, 140832, 142600, 143013, 146029, 147793, 151465, 151627, 153495, 154029, 154083, 155244, 158539, 158787, 160303, 162184, 162588, 163215, 163803, 164632
二十-十二面体图18087, 122, 127, 140, 147, 152, 157, 160
小斜方二十面截半十二面体图11484052543, 60383, 72548, 81253, 83903, 92075, 92185, 95313, 96068, 100983, 103003, 104443, 106023, 108848, 109713, 110795, 110905, 113423, 113653, 115208, 115823, 116623, 117180
小斜方截半立方八面体图1680767, 843, 1028, 1071, 1133, 1229, 1263, 1292, 1323, 1343, 1368
扭棱立方体图3801614137, 14316, 15137, 18995, 19063, 19248, 20069, 20143, 21661, 21803, 22068, 22099, 22691, 23023, 23171, 23244
扭棱十二面体图7171620026954193, 27485504, 29823985, 37376431, 38225816, 40564297, 40882371, 40985079, 44358325, 45182813, 45417384, 45660607, 45978681, 46559183, 48175213, 48958491, 49240567, 49914079, 49964316, 50687019, 50856597, 51341449, 52281493, 52553379, 52608385, 52759287, 52770720, 53258901, 53486365, 54026481, 54238007, 54360689, 54538180, 55029105, 55182621, 55349725, 55370172
截角立方体图6035, 45, 65, 77, 78, 80, 83, 87, 91, 93, 94
截角十二面体图450267, 351, 519, 635, 640, 672, 731, 751, 755, 788, 810, 835, 863, 876, 890, 896, 907, 915, 920, 934, 946, 952, 955
截角二十面体图2508016273, 16778, 23234, 24749, 27274, 29359, 29864, 31488, 32519, 33133, 33835, 34405, 34843, 35369, 36048, 36704, 36769, 37534, 37859, 38054, 38438, 38503, 38760
截角八面体图1008625, 682, 810, 981, 1081, 1096, 1153, 1197, 1242, 1258, 1273, 1296
截角四面体图3017, 21, 29, 32, 33

另请参阅

Foster 定理, 基尔霍夫指数, 基尔霍夫和指数, 电阻等价图, 电阻网络, 维纳和指数

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参考文献

Babić, D.; Klein, D. J.; Lukovits, I.; Nikolić, S.; 和 Trinajstić, N. "Resistance-Distance Matrix: A Computational Algorithm and Its Applications." Int. J. Quant. Chem. 90, 166-176, 2002.Devillers, J. 和 Balaban, A. T. (编). Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, pp. 81-82, 2000.Jeans, J. H. Chapter 9, Question 17 in The Mathematical Theory of Electricity and Magnetism, 5th ed. Cambridge, England: University Press, p. 337, 1925.Klein, D. J. 和 Randić, M. "Resistance Distance." J. Math. Chem. 12, 81-95, 1993.Klein, D. J. "Resistance-Distance Sum Rules." Croatica Chem. Acta 75, 633-649, 2002.Lukovits, I.; Nikolić, S.; 和 Trinajstić, N. "Resistance Distance in Regular Graphs." Int. J. Quan. Chem. 71, 217-225, 1999.Lukovits, I.; Nikolić, S.; 和 Trinajstić, N. "Note on the Resistance Distances in the Dodecahedron." Croatica Chem. Acta 73, 957-967, 2000.Palacios, J. L. "Closed-Form Formulas for Kirchhoff Index." Int. J. Quant. Chem. 81, 135-140, 2001.Xiao, W. 和 Gutman, I. "Resistance Distance and Laplacian Spectrum." Theor. Chem. Acc. 110, 284-289, 2003.

在 上被引用

电阻距离

引用为

Weisstein, Eric W. "电阻距离。" 来自 Web 资源。 https://mathworld.net.cn/ResistanceDistance.html

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