Q(L)-spectra of two join graphs and infinite families of Q(L)-integral graphs

Let G be a simple graph. The matrix Q(G) = D(G) + A(G) is the signless Laplacian matrix of G, where D(G) and A(G) is the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. The Laplacian matrix of G is the matrix L(G) = D(G)-A(G). A graph iscalled L-integral (resp. Q-integral) if its (signless) Laplacian spectrum consists entirely of integers. Let G ₁ and G₂ be two graphs, and S(G) be the subdivision graph of G. Then the S_vertex join of G₁ and G₂, denoted by G₁ V G₂ is obtained from S(G₁ ) and G₂ by joining each vertices of G₁ to each vertices of G₂. The S_edge join of G₁ with G₂, denoted by G₁ VG₂ is obtained from S(G₁) and G₂ by joining all vertices of S(G₁) corresponding to the edges of G₁ with all vertices of G₂. In this paper we obtain the Q-spectra and L-spectra of these two joins of graphs when G ₁ and G₂ are regular graphs. As an application, some infinite families of Q-integral graphs and L-integral graphs are obtained.