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
Clique
المؤلف:
Harary, F
المصدر:
Graph Theory. Reading, MA: Addison-Wesley, 1994.
الجزء والصفحة:
...
4-3-2022
3857
+
-
20
Clique
A clique of a graph 
is a complete subgraph of 
, and the clique of largest possible size is referred to as a maximum clique (which has size known as the clique number 
). However, care is needed since maximum cliques are often called simply "cliques" (e.g., Harary 1994). A maximal clique is a clique that cannot be extended by including one more adjacent vertex, meaning it is not a subset of a larger clique. Maximum cliques are therefore maximal cliqued (but not necessarily vice versa).
Cliques arise in a number of areas of graph theory and combinatorics, including graph coloring and the theory of error-correcting codes.
A clique of size 
is called a 
-clique (though this term is also sometimes used to mean a maximal set of vertices that are at a distance no greater than 
from each other). 0-cliques correspond to the empty set (sets of 0 vertices), 1-cliques correspond to vertices, 2-cliques to edges, and 3-cliques to 3-cycles.
The clique polynomial is of a graph 
is defined as
 |
where 
is the number of cliques of size 
, with 
, 
equal to the vertex count of 
, 
equal to the edge count of 
, etc.
In the Wolfram Language, the command FindClique[g][[1]] can be used to find a maximum clique, and FindClique[g, Length /@ FindClique[g], All] to find all maximum cliques. Similarly, FindClique[g, Infinity] can be used to find a maximal clique, and FindClique[g, Infinity, All] to find all maximal cliques. To find all cliques, enumerate all vertex subsets 
and select those for which CompleteGraphQ[g, s] is true.
In general, FindClique[g, n] can be used to find a maximal clique containing at least 
vertices, FindClique[g, n, All] to find all such cliques, FindClique[g, {" src="https://mathworld.wolfram.com/images/equations/Clique/Inline17.svg" style="height:22px; width:6px" />n
}" src="https://mathworld.wolfram.com/images/equations/Clique/Inline18.svg" style="height:22px; width:6px" />] to find a clique containing at exactly
vertices, and FindClique[g, {" src="https://mathworld.wolfram.com/images/equations/Clique/Inline20.svg" style="height:22px; width:6px" />n
}" src="https://mathworld.wolfram.com/images/equations/Clique/Inline21.svg" style="height:22px; width:6px" />, All] to find all such cliques.
The numbers of cliques, equal to the clique polynomial evaluated at 
, for various members of graph families are summarized in the table below, where the trivial 0-clique represented by the initial 1 in the clique polynomial is included in each count.
| graph family | OEIS | number of cliques |
| alternating group graph | A308599 | X, 2, 8, 45, 301, 2281, ... |
| Andrásfai graph | A115067 | 4, 11, 21, 34, 50, 69, 91, ... |
 antelope graph |
A308600 | 2, 5, 10, 17, 34, 61, 98, ... |
| antiprism graph | A017077 | X, X, 27, 33, 41, 49, 57, 65, ... |
| Apollonian network | A205248 | 16, 40, 112, 328, 976, 2920, ... |
| barbell graph | A000079 | X, X, 16, 32, 64, 128, 256, 512, ... |
 bishop graph |
A183156 | 2, 7, 22, 59, 142, 319, ... |
 black bishop graph |
A295909 | 2, 4, 14, 30, 82, 160, 386, ... |
book graph  |
A016897 | 9, 14, 19, 24, 29, 34, 39, 44, ... |
| Bruhat graph | A139149 | 2, 4, 13, 61, 361, 2521, 20161, ... |
| centipede graph | A008586 | 4, 8, 12, 16, 20, 24, 28, 32, 36, ... |
cocktail party graph  |
A000244 | 3, 9, 27, 81, 243, 729, 2187, ... |
complete graph  |
A000079 | 2, 4, 8, 16, 32, 64, 128, 256, ... |
complete bipartite graph  |
A000290 | 4, 9, 16, 25, 36, 49, 64, 81, 100, ... |
complete tripartite graph  |
A000578 | 8, 27, 64, 125, 216, 343, 512, ... |
 -crossed prism graph |
A017281 | X, 21, 31, 41, 51, 61, 71, ... |
crown graph  |
A002061 | X, X, 13, 21, 31, 43, 57, 73, 91, ... |
| cube-connected cycle graph | A295926 | X, X, 69, 161, 401, 961, 2241, 5121, ... |
cycle graph  |
A308602 | X, X, 8, 9, 11, 13, 15, 17, 19, ... |
| dipyramidal graph | A308603 | X, X, 24, 27, 33, 39, 45, 51, 57, 63, ... |
empty graph  |
A000027 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ... |
| Fibonacci cube graph | A291916 | 4, 6, 11, 19, 34, 60, 106, 186, ... |
 fiveleaper graph |
A308604 | X, 4, 16, 25, 57, 129, 289, 641, 1409, ... |
| folded cube graph | A295921 | 3, 15, 24, 56, ... |
| gear graph | A016873 | X, X, 17, 22, 27, 32, 37, 42, 47, 52, ... |
grid graph  |
A056105 | 2, 9, 22, 41, 66, 97, 134, 177, 226, 281, ... |
grid graph  |
A295907 | 2, 21, 82, 209, 426, 757, 1226, 1857, ... |
| halved cube graph | A295922 | 2, 4, 16, 81, 393, 1777, ... |
| Hanoi graph | A295911 | 8, 25, 76, 229, 688, ... |
| helm graph | A016933 | X, X, 22, 26, 32, 38, 44, 50, 56, ... |
hypercube graph  |
A132750 | 4, 9, 21, 49, 113, 257, 577, 1281, 2817, ... |
| Keller graph | A295902 | 5, 57, 14833, 2290312801, ... |
 king graph |
A295906 | 2, 16, 50, 104, 178, 272, 386, ... |
 knight graph |
A295905 | 2, 5, 18, 41, 74, 117, 170, 233, 306, 389, ... |
ladder graph  |
A016897 | 4, 9, 14, 19, 24, 29, 34, 39, 44, 49, 54, ... |
ladder rung graph  |
A016777 | 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, ... |
| Menger sponge graph | A292209 | 45, 1073, 22977, ... |
Möbius ladder  |
A016861 | X, X, 16, 21, 26, 31, 36, 41, 46, 51, ... |
| Mycielski graph | A199109 | 2, 4, 11, 32, 95, 284, 851, 2552, 7655, ... |
odd graph  |
A295934 | 2, 8, 26, 106, 442, 1849, 7723, ... |
| pan graph | A005408 | X, X, 10, 11, 13, 15, 17, 19, 21, 23, ... |
path graph  |
A005843 | 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ... |
path complement graph  |
A000045 | 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... |
| permutation star graph | A139149 | 2, 4, 13, 61, 361, 2521, ... |
| polygon diagonal intersection graph | A300524 | X, X, 8, 18, 41, 80, 169, 250, ... |
prism graph  |
A016861 | X, X, 18, 21, 26, 31, 36, 41, 46, 51, ... |
 queen graph |
A295903 | 2, 16, 94, 293, 742, 1642, 3458, 7087, ... |
rook graph  |
A288958 | 2, 9, 34, 105, 286, 721, 1730, ... |
rook complement graph  |
A002720 | 2, 7, 34, 209, 1546, 13327, 130922, ... |
| Sierpiński carpet graph | A295932 | 17, 153, 1289, 10521, ... |
| Sierpiński sieve graph | A295933 | 8, 20, 55, 160, 475, ... |
| Sierpiński tetrahedron graph | A292537 | 6, 59, 227, 899, 3587, 14339, ... |
star graph  |
A005843 | 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, ... |
| sun graph | A295904 | X, X, 20, 32, 52, 88, 156, 288, 548, ... |
sunlet graph  |
A016813 | X, X, 14, 17, 21, 25, 29, 33, 37, 41, 45, ... |
| tetrahedral graph | A289837 | X, X, X, X, X, 261, 757, 2003, 5035, ... |
torus grid graph  |
A056107 | X, X, 34, 49, 76, 109, 148, 193, ... |
| transposition graph | A308606 | 2, 4, 16, 97, 721, 6121, ... |
| triangular graph | A290056 | X, 2, 8, 27, 76, 192, 456, 1045, ... |
| triangular grid graph | A242658 | 8, 20, 38, 62, 92, 128, 170, 218, ... |
triangular snake graph  |
A016789 | 2, X, 8, X, 14, X, 20, X, 26, X, 32, X, ... |
| web graph | A016993 | X, X, 24, 29, 36, 43, 50, 57, 64, 71, 78, ... |
wheel graph  |
A308607 | X, X, X, 16, 18, 22, 26, 30, 34, 38, 42, 46, ... |
 white bishop graph |
A295910 | X, 4, 9, 30, 61, 160, 301, 71, ... |
Closed forms for some of these are summarized in the table below.
| graph family | number of cliques |
| Andrásfai graph |  |
| antiprism graph | |
book graph  |
 |
cocktail party graph  |
 |
complete bipartite graph  |
 |
complete graph  |
 |
complete tripartite graph  |
 |
 -crossed prism graph |
 |
cycle graph  |
|
empty graph  |
 |
| gear graph |  |
| helm graph | |
hypercube graph  |
 |
| ladder graph |  |
ladder rung graph  |
 |
Möbius ladder  |
 |
| pan graph | |
path graph  |
 |
prism graph  |
|
star graph  |
 |
| sun graph |  |
sunlet graph  |
|
| web graph | |
wheel graph  |
|
REFERENCES
Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994.
Pemmaraju, S. and Skiena, S. Computational Discrete Mathematics: Combinatorics and Graph Theory in Mathematica. Cambridge, England: Cambridge University Press, pp. 247-248, 2003.
Skiena, S. "Maximum Cliques." §5.6.1 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 215 and 217-218, 1990.
Skiena, S. S. "Clique and Independent Set" and "Clique." §6.2.3 and 8.5.1 in The Algorithm Design Manual. New York: Springer-Verlag, pp. 144 and 312-314, 1997.
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)
