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. A maximum clique (i.e., clique of largest size in a given graph) is therefore always maximal, but the converse does not hold.
Maximal cliques are important in graph theoretic applications, including graph coloring, fractional graph coloring, and computation of other graph properties such as intersection number.
The Bron-Kerbosch algorithm is an efficient method for finding all maximal cliques in a graph. Tomita et al. (2006) gave a depth-first search algorithm with pruning methods that is similar to the Bron-Kerbosch algorithm, and the Wolfram Language can be used to find all maximal cliques in a given graph using this algorithm via FindClique[g, Infinity, All].
A maximal independent vertex set of a graph 
is equivalent to a maximal clique on the graph complement 
.
The polynomial in 
whose coefficients of 
are the numbers of maximal cliques of size 
may be called the maximal
clique polynomial. The size of a smallest maximal clique may be termed the lower clique number and the size of a largest
maximal clique (which is equivalent to the size of a maximum
clique) the (upper) clique number.
