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Resistance Distance

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The resistance distance between vertices i and j of a graph G 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 Omega_(ij) be the resistance distance between vertices i and j in a connected graph G on n nodes, and define

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

where L is the Laplacian matrix of G and 1 is the unit n×n matrix. Then the resistance distance matrix is given by

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

where A^(-1) 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)].
(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]
(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].
(5)
ResistanceMatrixPlatonic

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).

soliddenominatorsorted resistance distances
cubical graph127, 9, 10
dodecahedral graph3019, 27, 32, 34, 35
icosahedral graph3011, 14, 15
octahedral graph125, 6
tetrahedral graph21
ResistanceMatrixArchimedean

Similarly, the resistance distances for the Archimedean solids are given below and illustrated graphically above.

soliddenominatorsorted resistance distances
cuboctahedral graph2411, 14, 15, 16
great rhombicosidodecahedral graph267514380166172084, 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 graph10296063859, 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 graph18087, 122, 127, 140, 147, 152, 157, 160
small rhombicosidodecahedral graph11484052543, 60383, 72548, 81253, 83903, 92075, 92185, 95313, 96068, 100983, 103003, 104443, 106023, 108848, 109713, 110795, 110905, 113423, 113653, 115208, 115823, 116623, 117180
small rhombicuboctahedral graph1680767, 843, 1028, 1071, 1133, 1229, 1263, 1292, 1323, 1343, 1368
snub cubical graph3801614137, 14316, 15137, 18995, 19063, 19248, 20069, 20143, 21661, 21803, 22068, 22099, 22691, 23023, 23171, 23244
snub dodecahedral graph7171620026954193, 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 graph6035, 45, 65, 77, 78, 80, 83, 87, 91, 93, 94
truncated dodecahedral graph450267, 351, 519, 635, 640, 672, 731, 751, 755, 788, 810, 835, 863, 876, 890, 896, 907, 915, 920, 934, 946, 952, 955
truncated icosahedral graph2508016273, 16778, 23234, 24749, 27274, 29359, 29864, 31488, 32519, 33133, 33835, 34405, 34843, 35369, 36048, 36704, 36769, 37534, 37859, 38054, 38438, 38503, 38760
truncated octahedral graph1008625, 682, 810, 981, 1081, 1096, 1153, 1197, 1242, 1258, 1273, 1296
truncated tetrahedral graph3017, 21, 29, 32, 33

See also

Foster's Theorems, Kirchhoff Index, Kirchhoff Sum Index, Resistance-Equivalent Graphs, Resistor Network, Wiener Sum Index

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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 Balaban, A. T. (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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Resistance Distance

Cite this as:

Weisstein, Eric W. "Resistance Distance." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ResistanceDistance.html

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