المرجع الألكتروني للمعلوماتية
القرآن الكريم وعلومه
العقائد الإسلامية
الفقه وأصوله 
الرجال والحديث 
سيرة الرسول وآله 
علوم اللغة العربية 
الأدب العربي 
الأسرة والمجتمع 
الأخلاق والأدعية 
التاريخ 
الادارة والاقتصاد 
علم الفيزياء 
علم الكيمياء 
علم الأحياء 
الرياضيات 
الهندسة المدنية 
الزراعة 
الجغرافية 
القانون 
اللغة الأنكليزية 
الأخبار 
الإعلام 
مكتبة الصور 
أقلام بمختلف الألوان 
إضاءات
مفتاح
أجوبة الإستفتاءات الشرعية
وثائقيات
تاريخ الرياضيات
الرياضيات في الحضارات المختلفة
الرياضيات المتقطعة
الجبر
الهندسة
المعادلات التفاضلية و التكاملية
التحليل
علماء الرياضيات | Kirchhoff Index |
 2430
 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 
Page and Part : ...|
Read More
Date: 7-4-2022
1432
Date: 28-2-2022
1375
Date: 8-5-2022
1533
|
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."
|
|
|
|
تفوقت في الاختبار على الجميع.. فاكهة "خارقة" في عالم التغذية
|
|
|
|
|
|
|
أمين عام أوبك: النفط الخام والغاز الطبيعي "هبة من الله"
|
|
|
|
|
|
|
قسم شؤون المعارف ينظم دورة عن آليات عمل الفهارس الفنية للموسوعات والكتب لملاكاته
|
|
|
