For a graph H with vertex set \(V(H) =\{1,2,...,k\}\) and a family of vertex disjoint graphs \(H_1, H_2,...,H_k\) , the H-join of \(H_1,H_2,\ldots ,H_k\) , denoted by \(H[H_1,H_2,\ldots ,H_k]\) , is a graph obtained from H by replacing \(i^{th}\) vertex of H by \(H_i\) , \(1\le i \le k\) , and replacing any edge \(\{l,j\}\) of H by the set of new edges \(\{\{u_{l},u_{j}\}: u_{l} \in V(H_l), u_{j} \in V(H_j)\}\) . In this paper, we establish a necessary and sufficient condition under which the H-join of Laplacian integral graphs is Laplacian integral. We also give some necessary and sufficient conditions for the graphs \(H_{n_i}\) , \(1 \le i \le 5\) , so that \(P_4[H_{n_1}, H_{n_2}, H_{n_3}, H_{n_4}]\) and \(P_5[H_{n_1}, H_{n_2}, H_{n_3}, H_{n_4}, H_{n_5}]\) are Laplacian integral. It is to be noted that these graphs produce infinite family of Laplacian integral graphs which are not cographs. Furthermore, we provide characterizations for the H-join and H-product of constructably Laplacian integral graphs to be constructably Laplacian integral. While proving the above results, we have also given characterizations for H-join and H-product of cographs to be a cograph.
