clique_number(G, sampler=None, lagrange=2.0, **sampler_args)[source]#

Returns the number of vertices in the maximum clique of a graph. A maximum clique is a clique of the largest possible size in a given graph. The clique number math:omega(G) of a graph G is the number of vertices in a maximum clique in G. The intersection number of G is the smallest number of cliques that together cover all edges of G.

This function works by finding the maximum independent set of the compliment graph of the given graph G which is equivalent to finding maximum clique. It defines a QUBO with ground states corresponding to a maximum weighted independent set and uses the sampler to sample from it.

  • G (NetworkX graph) – The graph on which to find a maximum clique.

  • sampler – A binary quadratic model sampler. A sampler is a process that samples from low energy states in models defined by an Ising equation or a Quadratic Unconstrained Binary Optimization Problem (QUBO). A sampler is expected to have a ‘sample_qubo’ and ‘sample_ising’ method. A sampler is expected to return an iterable of samples, in order of increasing energy. If no sampler is provided, one must be provided using the set_default_sampler function.

  • lagrange (optional (default 2)) – Lagrange parameter to weight constraints (no edges within set) versus objective (largest set possible).

  • sampler_args – Additional keyword parameters are passed to the sampler.


clique_nodes – List of nodes that form a maximum clique, as determined by the given sampler.

Return type:



Samplers by their nature may not return the optimal solution. This function does not attempt to confirm the quality of the returned sample.


Maximum Clique on Wikipedia