The Kirchhoff sum index 
is a graph index defined for a
finite connected 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. The index of each value list in the
table starts at 1. An X indicates that the family in that row is not defined for
the corresponding parameter value or has no value in the domain of connected
graphs. The crossed prism graph row uses
a cycle size of 
,
and the folded cube graph row starts at 
.
| graph class | OEIS |  ,  , ... |
| Andrásfai graph | 1, 7, 183/14, 2085/109, 177683/7059, 3526539/113017, ... | |
| antiprism graph | X, X, 23/4, 437/42, 181/11, 853/36, 291887/9048, ... | |
| Apollonian network | 3, 672/85, 18269/475, 1359691/5295, 13731955316/7104375, ... | |
cocktail
party graph  | X, 4, 23/4, 23/3, 77/8, 58/5, 163/12, 109/7, 281/16, ... | |
complete
bipartite graph  | 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  | 2, 23/4, 19/2, 53/4, 17, 83/4, ... | |
crossed
prism graph with cycle size  | X, 227/18, 117/4, 925/18, 7091/90, 5008/45, 9374/63, ... | |
| crown graph | X, X, 21/2, 227/18, 551/36, 363/20, 379/18, 3023/126, ... | |
| cube-connected cycle graph | X, X, 18037/150, 52720110713/81144000, ... | |
cycle graph  | X, X, 2, 4, 7, 21/2, 15, 20, 26, 65/2, 40, 48, 57, 133/2, ... | |
| folded cube graph | X, 1, 3, 10, 65/2, 869/9, 5327/18, 7996/9, 33277/12, ... | |
| gear graph | X, X, 229/20, 170/9, 6475/228, 7189/180, 45563/852, 69, ... | |
grid graph  | 0, 4, 170/9, 44299/840, 39633487/346500, 133000367/623700, ... | |
grid graph  | 0, 227/18, 1900321/18900, 3012514861/7197120, ... | |
| halved cube graph | 0, 1, 3, 23/3, 115/6, 13123/270, 5698/45, 21445/63, ... | |
hypercube graph  | 1, 4, 227/18, 334/9, 96661/900, 140663/450, 10194871/11025, ... | |
Möbius ladder  | X, X, 7, 183/14, 1999/99, 1145/39, 14600/369, 60311/1164, 2929/45, ... | |
| Mycielski graph | 0, 1, 7, 5935/278, 10873314919/187685230, ... | |
odd graph  | 0, 2, 21, 169, 5425/4, 69991/6, ... | |
| pan graph | X, X, 14/3, 89/12, 169/15, 1117/72, 125/6, 6379/240, 9013/270, ... | |
| path graph | 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, ... | |
permutation
star graph  | 0, 1, 21/2, 2173/20, 31688/21, 5223853/168, ... | |
prism graph  | X, X, 36/5, 227/18, 384/19, 3461/120, 16855/426, 32317/630, ... | |
rook graph  | 0, 4, 13, 57/2, 52, 85, 129, 371/2, ... | |
star
graph  | A000217 | 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, ... |
| sun graph | X, X, 95/12, 79/6, 7175/372, 20479/780, 84069/2468, ... | |
sunlet
graph  | X, X, 38/3, 137/6, 37, 1067/20, 737/10, 577/6, 3679/30, ... | |
| tetrahedral graph | X, X, X, X, X, 367/12, 799/12, 3475/27, 16471/72, 13741/36, ... | |
| triangular graph | X, 0, 2, 23/4, 93/8, 20, 125/4, 183/4, 511/8, 86, ... | |
| web graph | X, X, 116/5, 2863/72, 3536/57, 19601/225, 50081/426, 27107/180, ... | |
wheel
graph  | 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.
with 
with 
