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# Kirchhoff Sum Index

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.

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