The maximum clique problem is an NP-hard problem. If there is an algorithm that extracts the maximum clique in polynomial time, the two complexity classes P and NP are the same. We try to delete the vertices that do not belong to the maximum clique from the input graph G with vertex set V using eigenvalue relation. In this paper, we identify the vertices that do not belong to the maximum clique by controlling the change of the eigenvalue distribution of the adjacency matrix due to the vertex deletion. First, we create the graph G_s, which does not have the integer eigenvalue from G. Next, we prove the following; (1) Let k_m be the number of the eigenvalues of A_s smaller than -1. We keep k_m by adding edges with the weights of -1 between the vertices in V before and after deletion to keep the size of the maximum clique. (2) Suppose the input graph is not a complete graph and if k_m decreases for any vertex deletion. When we can add an edge with the weight -1 that increases k_m after deleting vertex v, it does not belong to the maximum clique. (3) Let k_q be the integer series that does not exceed the positive eigenvalues of the adjacent matrix of G_s. If k_q keeps before and after deletion, the vertex does not belong to the maximum clique. And then, we demonstrate an algorithm to extract the maximum clique in polynomial time using this result.