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علماء الرياضيات | Kirchhoff Sum Index |
 1793
 01:48 صباحاً
date: 7-4-2022 |
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. 90 
Page and Part : ...|
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Date: 28-7-2016
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Date: 10-3-2022
2014
Date: 2-3-2022
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The Kirchhoff sum index 
is a graph index defined for a graph on 
nodes by
 |
where 
is the resistance distance matrix and 
is the graph distance matrix.
Unless otherwise stated, hydrogen atoms are usually ignored in the computation of such indices as organic chemists usually do when they write a benzene ring as a hexagon (Devillers and Balaban 1999, p. 25).
Precomputed values for many graphs are implemented in the Wolfram Language as GraphData[g, "KirchhoffSumIndex"].
The following table summarizes values of the Kirchhoff sum index for various special classes of graphs.
| graph class | OEIS |  ,  , ... |
| Andrásfai graph | A000000 | 1, 7, 183/14, 2085/109, 177683/7059, 3526539/113017, ... |
| antiprism graph | A000000 | X, X, 23/4, 437/42, 181/11, 853/36, 291887/9048, ... |
| Apollonian network | A000000 | 3, 672/85, 18269/475, 1359691/5295, 13731955316/7104375, ... |
cocktail party graph  |
A000000 | X, 4, 23/4, 23/3, 77/8, 58/5, 163/12, 109/7, 281/16, ... |
complete bipartite graph  |
A000000 | 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, ... |
complete graph  |
A001477 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ... |
complete tripartite graph  |
A000000 | 2, 23/4, 19/2, 53/4, 17, 83/4, ... |
| crossed prism graph | A000000 | 227/18, 117/4, 925/18, 7091/90, 5008/45, 9374/63, ... |
| crown graph | A000000 | X, X, 21/2, 227/18, 551/36, 363/20, 379/18, 3023/126, ... |
| cube-connected cycle graph | A000000 | X, X, 18037/150, 52720110713/81144000, ... |
cycle graph  |
A000000 | X, X, 2, 4, 7, 21/2, 15, 20, 26, 65/2, 40, 48, 57, 133/2, ... |
| folded cube graph | A000000 | 1, 3, 10, 65/2, 869/9, 5327/18, 7996/9, 33277/12, ... |
| gear graph | A000000 | X, X, 229/20, 170/9, 6475/228, 7189/180, 45563/852, 69, ... |
grid graph  |
A000000 | 0, 4, 170/9, 44299/840, 39633487/346500, 133000367/623700, ... |
grid graph  |
A000000 | 0, 227/18, 1900321/18900, 3012514861/7197120, ... |
| halved cube graph | A000000 | 0, 1, 3, 23/3, 115/6, 13123/270, 5698/45, 21445/63, ... |
hypercube graph  |
A000000 | 1, 4, 227/18, 334/9, 96661/900, 140663/450, 10194871/11025, ... |
Möbius ladder  |
A000000 | 7, 183/14, 1999/99, 1145/39, 14600/369, 60311/1164, 2929/45, ... |
| Mycielski graph | A000000 | 0, 1, 7, 5935/278, 10873314919/187685230, ... |
odd graph  |
A000000 | 0, 2, 21, 169, 5425/4, 69991/6, ... |
| pan graph | A000000 | 14/3, 89/12, 169/15, 1117/72, 125/6, 6379/240, 9013/270, ... |
| path graph | A000000 | 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, ... |
permutation star graph  |
A000000 | 0, 1, 21/2, 2173/20, 31688/21, 5223853/168, ... |
prism graph  |
A000000 | X, X, 36/5, 227/18, 384/19, 3461/120, 16855/426, 32317/630, ... |
rook graph  |
A000000 | X, 4, 13, 57/2, 52, 85, 129, 371/2, ... |
star graph  |
A000290 | 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, ... |
| sun graph | A000000 | X, X, 95/12, 79/6, 7175/372, 20479/780, 84069/2468, ... |
sunlet graph  |
A000000 | X, X, 38/3, 137/6, 37, 1067/20, 737/10, 577/6, 3679/30, ... |
| tetrahedral graph | A000000 | X, X, X, X, X, 367/12, 799/12, 3475/27, 16471/72, 13741/36, ... |
| triangular graph | A000000 | X, 0, 2, 23/4, 93/8, 20, 125/4, 183/4, 511/8, 86, ... |
| web graph | A000000 | X, X, 116/5, 2863/72, 3536/57, 19601/225, 50081/426, 27107/180, ... |
wheel graph  |
A000000 | X, X, X, 3, 14/3, 75/11, 189/20, 364/29, 564/35, 765/38, 123/5, ... |
Closed forms are summarized in the following table.
| graph |  |
cocktail party graph  |
 |
complete bipartite graph  |
 |
complete graph  |
 |
complete tripartite graph  |
 |
| crown graph |  |
cycle graph  |
 |
path graph  |
 |
rook graph  |
 |
star graph  |
 |
| triangular graph |  |
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, 1999.
Sloane, N. J. A. Sequences A000217/M2535 and A001477 in "The On-Line Encyclopedia of Integer Sequences."
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