تاريخ الرياضيات
الاعداد و نظريتها
تاريخ التحليل
تار يخ الجبر
الهندسة و التبلوجي
الرياضيات في الحضارات المختلفة
العربية
اليونانية
البابلية
الصينية
المايا
المصرية
الهندية
الرياضيات المتقطعة
المنطق
اسس الرياضيات
فلسفة الرياضيات
مواضيع عامة في المنطق
الجبر
الجبر الخطي
الجبر المجرد
الجبر البولياني
مواضيع عامة في الجبر
الضبابية
نظرية المجموعات
نظرية الزمر
نظرية الحلقات والحقول
نظرية الاعداد
نظرية الفئات
حساب المتجهات
المتتاليات-المتسلسلات
المصفوفات و نظريتها
المثلثات
الهندسة
الهندسة المستوية
الهندسة غير المستوية
مواضيع عامة في الهندسة
التفاضل و التكامل
المعادلات التفاضلية و التكاملية
معادلات تفاضلية
معادلات تكاملية
مواضيع عامة في المعادلات
التحليل
التحليل العددي
التحليل العقدي
التحليل الدالي
مواضيع عامة في التحليل
التحليل الحقيقي
التبلوجيا
نظرية الالعاب
الاحتمالات و الاحصاء
نظرية التحكم
بحوث العمليات
نظرية الكم
الشفرات
الرياضيات التطبيقية
نظريات ومبرهنات
علماء الرياضيات
500AD
500-1499
1000to1499
1500to1599
1600to1649
1650to1699
1700to1749
1750to1779
1780to1799
1800to1819
1820to1829
1830to1839
1840to1849
1850to1859
1860to1864
1865to1869
1870to1874
1875to1879
1880to1884
1885to1889
1890to1894
1895to1899
1900to1904
1905to1909
1910to1914
1915to1919
1920to1924
1925to1929
1930to1939
1940to the present
علماء الرياضيات
الرياضيات في العلوم الاخرى
بحوث و اطاريح جامعية
هل تعلم
طرائق التدريس
الرياضيات العامة
نظرية البيان
Molecular Topological Index
المؤلف:
Balaban, A. T.; Motoc, I.; Bonchev, D.; and Mekenyan, O
المصدر:
"Topological Indices for Structure-Activity Correlations." Top. Curr. Chem. 114,
الجزء والصفحة:
...
7-4-2022
3848
+
-
20
Molecular Topological Index
The molecular topological index is a graph index defined by
 |
where 
are the components of the vector
 |
with 
the adjacency matrix, 
the graph distance matrix, and 
the vector of vertex degrees of a graph. The molecular topological index is well-defined only for connected graphs, being indeterminate for disconnected graphs having isolated nodes and infinity for all other disconnected graphs.
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 molecular topological index is not very discriminant, with the three 10-node nonisomorphic graphs illustrated above for example sharing the same index value of 440 (Devillers and Balaban 1999, p. 140). In fact, the paw graph and square graph on four nodes are already indistinguishable using the index (both have index 48), with the number of non-MTI-unique connected graphs on 
, 2, ... nodes given by 0, 0, 0, 2, 12, 87, 815, 11086, ... (OEIS A193125).
Precomputed values of the molecular topological index for common graphs are implemented in the Wolfram Language as GraphData[graph, "MolecularTopologicalIndex"].
The following table summarizes values of the molecular topological index for various special classes of graphs.
| graph class | OEIS |  ,  , ... |
| Andrásfai graph | A192790 | 4, 80, 336, 880, 1820, 3264, 5320, ... |
| antiprism graph | A192791 | X, X, 240, 448, 760, 1200, 1792, 2560, ... |
| Apollonian network | A192792 | 72, 360, 2556, 22572, 219636, 2204244, ... |
cocktail party graph  |
A181773 | X, 48, 240, 672, 1440, 2640, 4368, 6720, 9792, ... |
complete bipartite graph  |
A192418 | 4, 48, 180, 448, 900, 1584, 2548, 3840, 5508, ... |
complete graph  |
A181617 | 0, 4, 24, 72, 160, 300, 504, 784, 1152, ... |
complete tripartite graph  |
A192491 | 1, 10, 36, 88, 175, 306, 490, 736, ... |
| crossed prism graph | A192793 | X, 360, 900, 1872, 3420, 5688, 8820, ... |
| crown graph | A192796 | X, X, 132, 360, 760, 1380, 2268, 3472, 5040, ... |
| cube-connected cycle graph | A192191 | X, X, 5544, 57408, 458400, 3339648, 21641088, ... |
cycle graph  |
A192797 | X, X, 24, 48, 80, 132, 196, 288, ... |
| folded cube graph | A192826 | X, 72, 448, 2400, 13824, 72128, 389120, ... |
| gear graph | A192827 | X, X, 11, 88, 231, 440, 715, 1056, ... |
grid graph  |
A192828 | X, 48, 440, 2008, 6468, 16736, 37248, ... |
grid graph  |
A192829 | 360, 8064, 68928, 355470, 1340424, 4086180, ... |
| halved cube graph | A192830 | 0, 4, 72, 672, 4800, 30240, ... |
hypercube graph  |
A192831 | 4, 48, 360, 2304, 13600, 76032, 407680, ... |
Möbius ladder  |
A192833 | X, X, 180, 336, 600, 936, 1428, 2016, 2808, ... |
| Mycielski graph | A192834 | 0, 4, 80, 800, 6248, 43424, 283880, 1793600, ... |
odd graph  |
A192835 | 0, 24, 540, 12040, 258300, 5258484, ... |
| pan graph | A192836 | X, X, 14, 29, 48, 83, 126, 193, 272, 383, 510, ... |
path graph  |
A121318 | 0, 4, 16, 38, 74, 128, 204, 306, 438, 604, 808, ... |
permutation star graph  |
A192837 | 0, 4, 132, 4680, 214080, 12416400, ... |
prism graph  |
A192838 | X, X, 180, 360, 600, 972, 1428, 2064, 2808, ... |
rook graph  |
A192832 | X, 48, 576, 2880, 9600, 25200, 56448, 112896, ... |
star graph  |
A016742 | 0, 4, 16, 36, 64, 100, 144, 196, 256, ... |
| sun graph | A192845 | X, X, 180, 400, 740, 1224, 1876, 2720, 3780, ... |
sunlet graph  |
A192846 | X, X, 126, 256, 430, 696, 1022, 1472, ... |
| tetrahedral graph | A192847 | 7020, 30240, 100800, 281232, 687960 |
| triangular graph | A192849 | X, 0, 24, 240, 1080, 3360, 8400, 18144, ... |
| web graph | A192850 | X, X, 414, 832, 1390, 2232, 3262, 4672, |
wheel graph  |
A139098 | X, X, X, 72, 128, 200, 288, 392, 512, ... |
Closed forms are summarized in the following table.
| graph |  |
| Andrásfai graph |  |
| antiprism graph |  |
cocktail party graph  |
 |
complete bipartite graph  |
 |
complete graph  |
 |
complete tripartite graph  |
 |
| crossed prism graph |  |
| crown graph |  |
cycle graph  |
![1/4n[2n^2+(-1)^n+15]](https://mathworld.wolfram.com/images/equations/MolecularTopologicalIndex/Inline39.svg) |
| gear graph |  |
grid graph  |
 |
grid graph  |
 |
| Mycielski graph |  |
path graph  |
 |
prism graph  |
 |
| rook graph |  |
star graph  |
 |
| sun graph |  |
sunlet graph  |
![n[2n(n+3)+(-1)^n+7]](https://mathworld.wolfram.com/images/equations/MolecularTopologicalIndex/Inline55.svg) |
| triangular graph |  |
| web graph | ![n[6n^2+22n+3((-1)^n+7)]](https://mathworld.wolfram.com/images/equations/MolecularTopologicalIndex/Inline57.svg) |
wheel graph  |
 |
REFERENCES
Balaban, A. T.; Motoc, I.; Bonchev, D.; and Mekenyan, O. "Topological Indices for Structure-Activity Correlations." Top. Curr. Chem. 114, 21-55, 1983.
Devillers, J. and Balaban, A. T. (Eds.). Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, pp. 30-31, 138-141, and 210-212, 1999.
Mercader, E.; Castro, E. A.; and Toropov, A. A. "Maximum Topological Distances Based Indices as Molecular Descriptors for QSPR. 4. Modeling the Enthalpy of Formation of Hydrocarbons from Elements." Int. J. Mol. Sci. 2, 121-132, 2001.
Mueller, W. R.; Szymanski, K.; Knop, J. V.; and Trinajstić, N. "Molecular Topological Index." J. Chem. Inf. Comput. Sci. 30, 160-163, 1990.Randić, M. "In Search of Structural Invariants." J. Math. Chem. 9, 97-146, 1992.
Schultz, H. P. "Topological Organic Chemistry. 1. Graph Theory and Topological Indices of Alkanes." J. Chem. Inf. Comput. Sci. 29, 227-228, 1989.
Schultz, H. P.; Schultz, E. B.; and Schultz, T. P. "Topological Organic Chemistry. Part 2. Graph Theory, Matrix Determinants and Eigenvalues, and Topological Indices of Alkanes." J. Chem. Inf. Comput. Sci. 30, 27-29, 1990.
Sloane, N. J. A. Sequences A016742, A139098, A121318, A181617, A181773, A192191, A192418, A192491, A192790, A192791, A192792, A192793, A192796, A192797, A192826, A192827, A192828, A192829, A192830, A192831, A192832, A192833, A192834, A192835, A192836, A192837, A192838, A192839, A192845, A192846, A192847, A192848, A192849, A192850, and A193125 in "The On-Line Encyclopedia of Integer Sequences."
الاكثر قراءة في نظرية البيان
اخر الاخبار
اخبار العتبة العباسية المقدسة
الآخبار الصحية
قسم الشؤون الفكرية يصدر كتاباً يوثق تاريخ السدانة في العتبة العباسية المقدسة
"المهمة".. إصدار قصصي يوثّق القصص الفائزة في مسابقة فتوى الدفاع المقدسة للقصة القصيرة
(نوافذ).. إصدار أدبي يوثق القصص الفائزة في مسابقة الإمام العسكري (عليه السلام)
