Before revision:  In this paper, a generalized algorithm for the k-Clique problemhas been newly proposed using a classical-quantum hybrid approach , which can also be extended as a generalized algorithm for maximum clique problem .
After revision:  In this paper, apart from theoretical solution of such k-clique problem, practical quantum gate-based implementation has been addressed using Grover's algorithm. This approach is further extended to design circuit for the maximum clique problem .