The (upper) vertex independence number of a graph, often called simply "the" independence number, is the cardinality of the largest independent vertex set, i.e., the size of a maximum independent vertex set (which is the same as the size of a largest maximal independent vertex set). The independence number is most commonly denoted , but may also be written (e.g., Burger et al. 1997) or (e.g., Bollobás 1981).
The independence number of a graph is equal to the largest exponent in the graph's independence polynomial.
The lower independence number may be similarly defined as the size of a smallest maximal independent vertex set in (Burger et al. 1997).
The lower irredundance number , lower domination number , lower independence number , upper independence number , upper domination number , and upper irredundance number satsify the chain of inequalities
(1)

(Burger et al. 1997).
The ratio of the independence number of a graph to its vertex count is known as the independence ratio of (Bollobás 1981).
The independence number of a graph is equal to the clique number of the complement graph,
(2)

For a connected regular graph on vertices with vertex degree and smallest graph eigenvalue ,
(3)

(A. E. Brouwer, pers. comm., Dec. 17, 2012).
For the graph radius,
(4)

(DeLa Vina and Waller 2002). Lovasz (1979, p. 55) showed that when is the path covering number,
(5)

with equality for only complete graphs (DeLa Vina and Waller 2002).
The matching number of a graph is equal to the independence number of its line graph .
By definition,
(6)

where is the vertex cover number of and its vertex count (West 2000).
Known value for some classes of graph are summarized below.
graph  OEIS  values  
alternating group graph  A000000  1, 1, 4, 20, 120, ...  
Andrásfai graph ()  A000027  3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...  
antiprism graph ()  A004523  2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, ...  
Apollonian network  A000244  1, 3, 9, 27, 81, 243, 729, 2187, ...  
complete bipartite graph  A000027  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...  
complete graph  1  A000012  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 
complete tripartite graph  A000027  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...  
cycle graph ()  A004526  1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, ...  
empty graph  A000027  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...  
folded cube graph ()  A058622  1, 1, 4, 5, 16, 22, 64, 93, 256, ...  
grid graph  A000982  1, 2, 5, 8, 13, 18, 25, 32, 41, 50, 61, 72, ...  
grid graph  A036486  1, 4, 14, 32, 63, 108, 172, 256, 365, 500, ...  
halved cube graph  A005864  1, 1, 4, 5, 16, 22, 64, 93, 256, ...  
Hanoi graph  A000244  1, 3, 9, 27, 81, 243, 729, 2187, ...  
hypercube graph  A000079  1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...  
Keller graph  A258935  4, 5, 8, 16, 32, 64, 128, 256, 512, ...  
king graph ()  A008794  1, 4, 4, 9, 9, 16, 16, 25, 25  
knight graph ()  A030978  4, 5, 8, 13, 18, 25, 32, 41, 50, 61, 72, ...  
Kneser graph  
Mycielski graph  A266550  1, 1, 2, 5, 11, 23, 47, 95, 191, 383, 767, ...  
Möbius ladder ()  A109613  3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, ...  
odd graph  A000000  1, 1, 4, 15, 56, 210, 792, 3003, 11440, ...  
pan graph  A000000  2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, ...  
path graph  A004526  1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...  
prism graph ()  A052928  2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, ...  
Sierpiński carpet graph  4, 32, 256, ...  
Sierpiński gasket graph  1, 3, 6, 15, 42, ...  
star graph  A028310  1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...  
triangular graph ()  A004526  1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, ...  
web graph ()  A032766  4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, ...  
wheel graph  A004526  1, 2, 2, 3, 3, 4, 4, 5, 5, ... 
Precomputed independence numbers for many named graphs can be obtained in the Wolfram Language using GraphData[graph, "IndependenceNumber"].