A coclique (resp. clique) extension of a graph H is a graph G obtained from H by replacing each vertex of H by a coclique (resp. clique), where vertices of G coming from different vertices of H are adjacent if and only if the original vertices are adjacent in H. Let \(\mathcal {M}_n(H)\) and \(\mathcal {M}^c_n(H)\) be the sets of graphs with order n, which are the coclique and clique extensions of H, respectively. In this paper, we discuss the minimum spectral radius in \(\mathcal {M}_n(H)\) and \(\mathcal {M}^c_n(H)\) with \(H\cong P_k\) or \(H\cong C_k\) , where \(P_k\) and \(C_k\) are the path of order k and the cycle of order k, respectively. We determine the graphs with the minimum spectral radius in \(\mathcal {M}_n(P_7)\) , \(\mathcal {M}^c_n(P_k)\) with \(k=3,\,4,\,n-2\) and \(\mathcal {M}_n(C_{n-2})\) , respectively.
