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علماء الرياضيات | Clique |
 2406
 01:58 صباحاً
date: 4-3-2022 |
Author : Harary, F 
Book or Source : Graph Theory. Reading, MA: Addison-Wesley, 1994. 
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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, 
n
] to find a clique containing at exactly 
vertices, and FindClique[g, 
n
, 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  |
|
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.
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