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علماء الرياضيات | Resistance Distance |
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 04:36 مساءً
date: 13-10-2021 |
Author : Babić, D.; Klein, D. J.; Lukovits, I.; Nikolić, S.; and Trinajstić, N 
Book or Source : "Resistance-Distance Matrix: A Computational Algorithm and Its Applications." Int. J. Quant. Chem 
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The resistance distance between vertices 
and 
of a graph 
is defined as the effective resistance between the two vertices (as when a battery is attached across them) when each graph edge is replaced by a unit resistor (Klein and Randić 1993, Klein 2002). This resistance distance is a metric on graphs (Klein 2002).
Let 
be the resistance distance between vertices 
and 
in a connected graph 
on 
nodes, and define
 |
(1) |
where 
is the Laplacian matrix of 
and 
is the unit 
matrix. Then the resistance distance matrix is given by
 |
(2) |
where 
denotes a matrix inverse (Babić et al. 2002). This can be written explicitly as
![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)].](https://mathworld.wolfram.com/images/equations/ResistanceDistance/NumberedEquation3.gif) |
(3) |
Graphs that have identical resistance distance sets are known as resistance-equivalent graphs. The smallest such pairs of graphs have nine vertices.
For example, the resistance distance matrix for the tetrahedral graph is
![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]](https://mathworld.wolfram.com/images/equations/ResistanceDistance/NumberedEquation4.gif) |
(4) |
and for the cubical graph is given by
![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].](https://mathworld.wolfram.com/images/equations/ResistanceDistance/NumberedEquation5.gif) |
(5) |
The resistance distances for the Platonic graphs (Klein 2002) are summarized in the following table, expressed over a common denominator, and illustrated graphically above. The case of the dodecahedral graph was considered by Jeans (1925).
| solid | denominator | sorted resistance distances |
| cubical graph | 12 | 7, 9, 10 |
| dodecahedral graph | 30 | 19, 27, 32, 34, 35 |
| icosahedral graph | 30 | 11, 14, 15 |
| octahedral graph | 12 | 5, 6 |
| tetrahedral graph | 2 | 1 |
Similarly, the resistance distances for the Archimedean solids are given below and illustrated graphically above.
| solid | denominator | sorted resistance distances |
| cuboctahedral graph | 24 | 11, 14, 15, 16 |
| great rhombicosidodecahedral graph | 267514380 | 166172084, 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 |
| great rhombicuboctahedral graph | 102960 | 63859, 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 |
| icosidodecahedral graph | 180 | 87, 122, 127, 140, 147, 152, 157, 160 |
| small rhombicosidodecahedral graph | 114840 | 52543, 60383, 72548, 81253, 83903, 92075, 92185, 95313, 96068, 100983, 103003, 104443, 106023, 108848, 109713, 110795, 110905, 113423, 113653, 115208, 115823, 116623, 117180 |
| small rhombicuboctahedral graph | 1680 | 767, 843, 1028, 1071, 1133, 1229, 1263, 1292, 1323, 1343, 1368 |
| snub cubical graph | 38016 | 14137, 14316, 15137, 18995, 19063, 19248, 20069, 20143, 21661, 21803, 22068, 22099, 22691, 23023, 23171, 23244 |
| snub dodecahedral graph | 71716200 | 26954193, 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 |
| truncated cubical graph | 60 | 35, 45, 65, 77, 78, 80, 83, 87, 91, 93, 94 |
| truncated dodecahedral graph | 450 | 267, 351, 519, 635, 640, 672, 731, 751, 755, 788, 810, 835, 863, 876, 890, 896, 907, 915, 920, 934, 946, 952, 955 |
| truncated icosahedral graph | 25080 | 16273, 16778, 23234, 24749, 27274, 29359, 29864, 31488, 32519, 33133, 33835, 34405, 34843, 35369, 36048, 36704, 36769, 37534, 37859, 38054, 38438, 38503, 38760 |
| truncated octahedral graph | 1008 | 625, 682, 810, 981, 1081, 1096, 1153, 1197, 1242, 1258, 1273, 1296 |
| truncated tetrahedral graph | 30 | 17, 21, 29, 32, 33 |
REFERENCES:
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.
Devillers, J. and A. T. Balaban (Eds.). 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. and 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.; and Trinajstić, N. "Resistance Distance in Regular Graphs." Int. J. Quan. Chem. 71, 217-225, 1999.
Lukovits, I.; Nikolić, S.; and 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. and Gutman, I. "Resistance Distance and Laplacian Spectrum." Theor. Chem. Acc. 110, 284-289, 2003.
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