المرجع الألكتروني للمعلوماتية
القرآن الكريم وعلومه
العقائد الإسلامية
الفقه وأصوله 
الرجال والحديث 
سيرة الرسول وآله 
علوم اللغة العربية 
الأدب العربي 
الأسرة والمجتمع 
الأخلاق والأدعية 
التاريخ 
الادارة والاقتصاد 
علم الفيزياء 
علم الكيمياء 
علم الأحياء 
الرياضيات 
الهندسة المدنية 
الزراعة 
الجغرافية 
القانون 
اللغة الأنكليزية 
الأخبار 
الإعلام 
مكتبة الصور 
أقلام بمختلف الألوان 
إضاءات
مفتاح
أجوبة الإستفتاءات الشرعية
وثائقيات
تاريخ الرياضيات
الرياضيات في الحضارات المختلفة
الرياضيات المتقطعة
الجبر
الهندسة
المعادلات التفاضلية و التكاملية
التحليل
علماء الرياضيات | Resistance Distance |
 1833
 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 
Page and Part : ...|
Read More
Date: 16-2-2016
974
Date: 26-9-2021
2104
Date: 11-10-2021
971
|
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.
|
|
|
|
دراسة يابانية لتقليل مخاطر أمراض المواليد منخفضي الوزن
|
|
|
|
|
|
|
اكتشاف أكبر مرجان في العالم قبالة سواحل جزر سليمان
|
|
|
|
|
|
|
اتحاد كليات الطب الملكية البريطانية يشيد بالمستوى العلمي لطلبة جامعة العميد وبيئتها التعليمية
|
|
|
