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.

graph
cocktail party graph
complete bipartite graph
complete graph
complete tripartite graph
crossed prism graph
crown graph
cycle graph  with
empty graph	{0 for n=1; infty otherwise" src="https://mathworld.wolfram.com/images/equations/KirchhoffIndex/Inline43.svg" style="height:61px; width:125px" />
hypercube graph
pan graph
path graph
rook graph
star graph
triangular graph

---

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."

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