On spectra of square edge-corona graphs

Given two connected graphs, G and H. The Order and size of Gand H, are u and v, and m and n, respectively. Graph G square edge-corona H, denoted by G ⋄² H and defined as G ⋄² H: = (G ⋄ H) ⋄ H, that is a graph obtained from G, the first v-copies of H that is H_h, with h ∈ {1,2,.., v], and the second v(1 + 2n + m)-copies of H that is H_hk, with h ∈ {1,2,.., v] and k ∈ {1,2,..,1 + 2n + m] and joining the terminal vertices of e_h ∈ E(G), with e_h = i_hj_h and i_h, j_h ∈ V(G), to all vertices of H_h, and then joining the terminal vertices of e_k = r_ks_k, e_k ∈ G ⋄ H, to all vertices of H_hk . In this paper, the spectrum of the adjacency, Laplacian, and signless-Laplacian matrices of the graph G ⋄² H will be analyzed further.