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علماء الرياضيات | Stability Index |
 1653
 02:29 صباحاً
date: 8-4-2022 |
Author : Devillers, J. and Balaban, A. T. 
Book or Source : opological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach 
Page and Part : pp. 27-28 and 105|
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Date: 24-2-2022
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Date: 9-3-2022
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Date: 6-8-2016
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The stability index 
of a graph 
is defined by
 |
where 
is the 
th coefficient of the characteristic polynomial and 
denotes the floor function.
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).
The following table summarizes values of the Kirchhoff sum index for various special classes of graphs.
| graph class | OEIS |  ,  , ... |
| Andrásfai graph | A000000 | 2, 2, 70, 312, 2176, 16054, 69882, ... |
| antiprism graph | A000000 | X, X, 13, 117, 606, 1453, 5013, 26141, 91867, ... |
| Apollonian network | A000000 | 10, 56, 11757, 108238346932, ... |
cocktail party graph  |
A000000 | 1, 5, 13, 73, 281, 1101, 4005, 14225, 49201, ... |
complete bipartite graph  |
A000000 | 2, 5, 10, 17, 26, 37, 50, 65, 82, 10, ... |
complete graph  |
A001477 | 0, 2, 2, 10, 24, 66, 160, 386, 896, 2050, ... |
complete tripartite graph  |
A000000 | 2, 13, 54, 49, 250, 109, ... |
| crossed prism graph | A000000 | 80, 1280, 11520, 128000, 1310720, 13844480, ... |
| crown graph | A000000 | X, X, 20, 80, 272, 832, 2368, 6400, 16640, ... |
| cube-connected cycle graph | A000000 | X, X, 702672, 2312110080000000, ... |
cycle graph  |
A000000 | X, X, 2, 5, 2, 20, 2, 45, 2, 125, 2, 320, 2, ... |
empty graph  |
A000000 | 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ... |
| folded cube graph | A000000 | 2, 10, 17, 150806, 1129150390625, ... |
grid graph  |
A000000 | 0, 5, 0, 4176, 0, 390590941, 0, 4363974545574685, ... |
grid graph  |
A000000 | 0, 80, 0, 473138263916015625, 0, ... |
| halved cube graph | A000000 | 0, 2, 10, 73, 1716177, 990952398727038, ... |
hypercube graph  |
A000000 | 2, 5, 80, 10625, 2662400000, 27254945465087890625, ... |
Möbius ladder  |
A000000 | X, X, 10, 70, 250, 580, 1690, 4870, 13690, 40650, 118810, ... |
| Mycielski graph | A000000 | 0, 2, 2, 432, 2440488, 113920311278592, ... |
odd graph  |
A000000 | 0, 2, 424, 46757736448, ... |
| pan graph | A000000 | X, X, 6, 0, 16, 0, 42, 0, 110, 0, ... |
| path graph | A000000 | 0, 2, 0, 5, 0, 13, 0, 34, 0, 89, 0, 233, ... |
permutation star graph  |
A000000 | 0, 2, 20, 1250000, ... |
prism graph  |
A000000 | X, X, 22, 80, 204, 500, 1684, 5120, 14062, ... |
rook graph  |
A000000 | 5, 112, 492593, 77149243008, 254773522981613013, ... |
star graph  |
A000290 | 0, 2, 0, 4, 0, 6, 0, 8, 0, 10, 0, ... |
| sun graph | A000000 | X, X, 23, 52, 115, 202, 761, 4390, 20771, ... |
sunlet graph  |
A000000 | X, X, 14, 32, 82, 200, 478, 1152, 2786, 6728, ... |
| tetrahedral graph | A000000 | X, X, X, X, X, 88567324, 6545276817256, ... |
| triangular graph | A000000 | X, 0, 2, 13, 748, 362144, 2185830840, ... |
| web graph | A000000 | X, X, 333, 56, 6253, 566, 120213, 2982, 2202832, ... |
wheel graph  |
A000000 | X, X, X, 10, 8, 26, 48, 78, 144, 220, 500, 650, ... |
Closed forms are summarized in the following table, where 
is a Fibonacci number.
| graph | stability index |
cocktail party graph  |
 |
complete bipartite graph  |
 |
complete graph  |
 |
| crossed prism graph |  |
| crown graph |  |
cycle graph  |
 |
empty graph  |
 |
| gear graph | 0 |
| pan graph |  |
path graph  |
 |
star graph  |
 |
sunlet graph  |
 |
Devillers, J. and Balaban, A. T. (Eds.). Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, pp. 27-28 and 105, 1999.
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