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.

graph
cocktail party graph
complete bipartite graph
complete graph
complete tripartite graph
crown graph
cycle 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.

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."