Resistance Distance

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

(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

(4)

and for the cubical graph is given by

(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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