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Kirchhoff Sum Index


The Kirchhoff sum index KfS is a graph index defined for a graph on n nodes by

 KfS=1/2sum_(i=1)^nsum_(j=1)^n((Omega)_(ij))/((d)_(ij)),

where (Omega)_(ij) is the resistance distance matrix and (d)_(ij) 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 classOEISKfS(G_1), KfS(G_2), ...
Andrásfai graphA0000001, 7, 183/14, 2085/109, 177683/7059, 3526539/113017, ...
antiprism graphA000000X, X, 23/4, 437/42, 181/11, 853/36, 291887/9048, ...
Apollonian networkA0000003, 672/85, 18269/475, 1359691/5295, 13731955316/7104375, ...
cocktail party graph K_(n×2)A000000X, 4, 23/4, 23/3, 77/8, 58/5, 163/12, 109/7, 281/16, ...
complete bipartite graph K_(n,n)A0000001, 4, 7, 10, 13, 16, 19, 22, 25, 28, ...
complete graph K_nA0014770, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
complete tripartite graph K_(n,n,n)A0000002, 23/4, 19/2, 53/4, 17, 83/4, ...
crossed prism graphA000000227/18, 117/4, 925/18, 7091/90, 5008/45, 9374/63, ...
crown graphA000000X, X, 21/2, 227/18, 551/36, 363/20, 379/18, 3023/126, ...
cube-connected cycle graphA000000X, X, 18037/150, 52720110713/81144000, ...
cycle graph C_nA000000X, X, 2, 4, 7, 21/2, 15, 20, 26, 65/2, 40, 48, 57, 133/2, ...
folded cube graphA0000001, 3, 10, 65/2, 869/9, 5327/18, 7996/9, 33277/12, ...
gear graphA000000X, X, 229/20, 170/9, 6475/228, 7189/180, 45563/852, 69, ...
grid graph P_n square P_nA0000000, 4, 170/9, 44299/840, 39633487/346500, 133000367/623700, ...
grid graph P_n square P_n square P_nA0000000, 227/18, 1900321/18900, 3012514861/7197120, ...
halved cube graphA0000000, 1, 3, 23/3, 115/6, 13123/270, 5698/45, 21445/63, ...
hypercube graph Q_nA0000001, 4, 227/18, 334/9, 96661/900, 140663/450, 10194871/11025, ...
Möbius ladder M_nA0000007, 183/14, 1999/99, 1145/39, 14600/369, 60311/1164, 2929/45, ...
Mycielski graphA0000000, 1, 7, 5935/278, 10873314919/187685230, ...
odd graph O_nA0000000, 2, 21, 169, 5425/4, 69991/6, ...
pan graphA00000014/3, 89/12, 169/15, 1117/72, 125/6, 6379/240, 9013/270, ...
path graphA0000000, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, ...
permutation star graph PS_nA0000000, 1, 21/2, 2173/20, 31688/21, 5223853/168, ...
prism graph Y_nA000000X, X, 36/5, 227/18, 384/19, 3461/120, 16855/426, 32317/630, ...
rook graph K_n square K_nA000000X, 4, 13, 57/2, 52, 85, 129, 371/2, ...
star graph S_nA0002900, 1, 3, 6, 10, 15, 21, 28, 36, 45, ...
sun graphA000000X, X, 95/12, 79/6, 7175/372, 20479/780, 84069/2468, ...
sunlet graph C_n circledot K_1A000000X, X, 38/3, 137/6, 37, 1067/20, 737/10, 577/6, 3679/30, ...
tetrahedral graphA000000X, X, X, X, X, 367/12, 799/12, 3475/27, 16471/72, 13741/36, ...
triangular graphA000000X, 0, 2, 23/4, 93/8, 20, 125/4, 183/4, 511/8, 86, ...
web graphA000000X, X, 116/5, 2863/72, 3536/57, 19601/225, 50081/426, 27107/180, ...
wheel graph W_nA000000X, 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.


See also

Balaban Index, Graph Distance Matrix, Kirchhoff Index, Resistance Distance

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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, 1999.Sloane, N. J. A. Sequences A000217/M2535 and A001477 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Kirchhoff Sum Index

Cite this as:

Weisstein, Eric W. "Kirchhoff Sum Index." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KirchhoffSumIndex.html

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