The (upper) clique number of a graph 
, denoted 
, is the number of vertices in a maximum
clique of 
.
Equivalently, it is the size of a largest clique or maximal clique of 
.
The clique number 
of a graph is equal to the largest exponent in the graph's clique
polynomial.
The lower clique number 
may be similarly defined as the size of a graph's
smallest maximal clique.
For an arbitrary graph,
 |
(1)
|
where 
is the vertex degree of 
.
The clique number of a graph is equal to the independence number of the complement graph,
 |
(2)
|
The chromatic number 
of a graph 
is equal to or greater than its clique number 
, i.e.,
 |
(3)
|
The following table lists the clique numbers for some named graphs.
The following table gives the number 
of 
-node graphs having clique number 
for small 
.
 | OEIS |  |
| 1 | 1, 1, 1, 1, 1, 1, 1, 1, ... | |
| 2 | A052450 | 0, 1, 2, 6, 13, 37, 106, 409, 1896, ... |
| 3 | A052451 | 0, 0, 1, 3, 15, 82, 578, 6021, 101267, ... |
| 4 | A052452 | 0, 0, 0, 1, 4, 30, 301, 4985, 142276, ... |
| 5 | A077392 | 0, 0, 0, 0, 1, 5, 51, 842, 27107, ... |
| 6 | A077393 | 0, 0, 0, 0, 0, 1, 6, 80, 1995, ... |
| 7 | A077394 | 0, 0, 0, 0, 0, 0, 1, 7, 117, ... |
| 8 | 0, 0, 0, 0, 0, 0, 0, 1, 8, ... |
." Acta Math. 182, 105-142,
1999.Sloane, N. J. A. Sequences