A coclique extension of a graph H is a graph G obtained from H by replacing each vertex of H by a coclique, where vertices of G coming from different vertices of H are adjacent if and only if the original vertices are adjacent in H. Let M n (H) be the set of graphs with order n, which are the coclique extensions of H. In this paper, we discuss the minimum spectral radius in M n (P k) and the maximum spectral radius in M n (C k), where P k and C k are the path of order k and the cycle of order k, respectively. Then we disprove a conjecture on the minimum spectral radius in M n (P k) and confirm a conjecture on the maximum spectral radius in M n (C k), which are given by Monsalve and Rada (2021)[4].