Abstract: Finding maximal clique of a graph is a fundamental problem in graph theory．By combining discernibility function with maximal clique,the discernibility function of maximal clique over the vertex and the Boolean mapping function of the vertex with respect to any vertex set were defined,and some characteristics and theorems relative to maximal clique were obtained,then a fact was proved that the problem of finding all maximal cliques in a graph can be naturally expressed as relatively simple constraints of discernibility function expression．Furthermore,discernibility function-based algorithm was designed for finding all maximal cliques in a graph by introducing the method of reduction tree constructing．The theoretical derivation and simulation instance show that this algorithm is feasible and effective in practice.

Key words: Graph theory,Maximal clique,Discernibility function,Reduction tree constructing