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# Kirchhoff Index

The Kirchhoff index , also simply called the resistance and denoted (Lukovits et al. 1999), of a connected graph on nodes is defined by

where is the resistance 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, "KirchhoffIndex"].

The following table summarizes values of the Kirchhoff index for various special classes of graphs.

 graph class OEIS , , ... Andrásfai graph A000000/A000000 1, 10, 134/7, 3080/109, 263599/7059, 5244806/113017, ... antiprism graph A000000/A000000 X, X, 13/2, 290/21, 551/22, 41, ... Apollonian network A000000/A000000 3, 834/85, 30154/475, 23555722/44125, 1259601793/263125, ... cocktail party graph A000000/A000000 , 5, 13/2, 25/3, 41/4, 61/5, 85/6, 113/7, 145/8, 181/9, ... complete bipartite graph A000000 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, ... complete graph A001477 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ... complete tripartite graph A000000 2, 13/2, 11, 31/2, 20, 49/2, ... crossed prism graph A000000 X, 58/3, 53, 332/3, 595/3, 322, 1463/3, 2104/3, 969, 3890/3, ... crown graph A000000/A000000 X, X, 35/2, 58/3, 271/12, 131/5, 899/30, 710/21, ... cube-connected cycle graph A000000/A000000 X, X, 346, 6442544/2415, 33760334655251468654052195628373/1958536428315956487415847430, ... cycle graph A138190/A138191 X, X, 2, 5, 10, 35/2, 28, 42, ... folded cube graph A000000/A000000 X, 1, 3, 13, 50, 548/3, 1960/3, 6968/3, 8272, ... gear graph A000000/A000000 X, X, 18, 69/2, 1085/19, 257/3, 8526/71, 6733/42, 10935/53, 53945/209, ... grid graph A000000/A000000 0, 5, 69/2, 884/7, 11155/33, 2520, 5488, 10752, ... grid graph A000000/A000000 0, 58/3, 16959/70, 489296/357, 119593165825/23110593, 136080, ... halved cube graph A290365/A290366 0, 1, 3, 25/3, 70/3, 3014/45, 2968/15, 63148/105, 197072/105, ... hypercube graph A290343/A290344 1, 5, 58/3, 206/3, 3548/15, ... Möbius ladder A000000/A000000 X, X, 9, 134/7, 1135/33, 725/13, 10367/123, 11732/97, 2835/17, ... Mycielski graph A000000/A000000 0, 1, 10, 4545/139, 8808777389/93842615, ... odd graph A000000/A000000 0, 2, 33, 373, 4000, 264001/6, ... pan graph A000000/A000000 X, X, 19/3, 23/2, 19, 88/3, 43, ... path graph A000000 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, ... permutation star graph A000000/A000000 0, 1, 35/2, 1314/5, 26520, 963598/7, ... prism graph A000000/A000000 X, X, 47/5, 58/3, 655/19, 279/5, ... rook graph A000000/A000000 X, 5, 18, 42, 80, 135, 210, 308, 432, ... star graph A000290 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, ... sun graph A000000/A000000 X, X, 65/6, 302/15, 75, 114, 161, 216, 279, 350, 429, ... sunlet graph A000000/A000000 X, X, 23, 48, 105, 174, 259, 376, 513, 690, 891, ... tetrahedral graph A000000/A000000 X, X, X, X, X, 43, 207/2, 1945/9, 4901/12, 2149/3, ... triangular graph A000000/A000000 X, 0, 2, 13/2, 57/4, 26, 85/2, 129/2, 371/4, 128, ... web graph A000000/A000000 X, X, 222/5, 173/2, 2780/19, 4521/20, 23282/71, 3179/7, 160848/265, ... wheel graph A000000/A000000 X, X, X, 16/3, 95/11, 129/10, ...

Closed forms are summarized in the following table. The cycle graph was considered by Klein et al. (1995) and Babić et al. (2002). Here, is a harmonic number and is the Lerch transcendent.

Balaban Index, Graph Distance Matrix, Kirchhoff Sum Index, Resistance Distance, Wiener Index, 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.Bonchev, D.; Balaban, A. T.; Liu, X.; and Klein, D. J. "Molecular Cyclicity and Centricity of Polycyclic Graphs. I. Cyclicity Based on Resistance Distances or Reciprocal Distances." Int. J. Quan. Chem. 50, 1-20, 1994.Devillers, J. and Balaban, A. T. (Eds.). Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, pp. 40-41 and 114, 1999.Klein, D. J.; Lukovits, I.; and Gutman, I. "On the Definition of the Hyper-Wiener Index for Cycle-Containing Structures." J. Chem. Inf. Comput. 35, 50-52, 1995.Lukovits, I.; Nikolić, S.; and Trinajstić, N. "Resistance Distance in Regular Graphs." Int. J. Quan. Chem. 71, 217-225, 1999.Palacios, J. L. "Closed-Form Formulas for Kirchhoff Index." Int. J. Quant. Chem. 81, 135-140, 2001.Sloane, N. J. A. Sequences A000290/M3356, A001477, A138190, A138191, A290343, A290344, A290347, A290348, A290365, and A290366 in "The On-Line Encyclopedia of Integer Sequences."

Kirchhoff Index

## Cite this as:

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