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علماء الرياضيات | Kirchhoff Index |
 2372
 04:11 مساءً
date: 7-4-2022 |
Author : Babić, D.; Klein, D. J.; Lukovits, I.; Nikolić, S.; and Trinajstić, N. 
Book or Source : "Resistance-Distance Matrix: A Computational Algorithm and Its Applications." Int. J. Quant. Chem. 90 
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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  |
 |
hypercube graph  |
![-2^(n-1)H_n-4^nR[Phi(z,1,n+1)]](https://mathworld.wolfram.com/images/equations/KirchhoffIndex/Inline45.svg) |
| pan graph |  |
path graph  |
 |
rook graph  |
 |
star graph  |
 |
| triangular graph |  |
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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