Some new families of integral graphs

Let G i , i = 1,2,3 be finite simple graphs with vertex sets $$V\left( {{G_1}} \right)\, = \,\left\{ {{u_1},{u_2},...,{u_{{n_1}}}} \right\},\,V\left( {{G_2}} \right)\, = \,\left\{ {{v_1},{v_2},...,{v_{{n_2}}}} \right\}\;and\;\,V\left( {{G_3}} \right)\, = \,\left\{ {{w_1},{w_2},...,{w_{{n_3}}}} \right\}\;$$ . In this paper, for each ordered triple of graphs (G 1, G 2, G 3), we define a new composition ψ(G 1, G 2, G 3) : The vertex set of ψ(G 1, G 2, G 3) is (V(G 1) ⨄ V(G 2)) × V(G 3) and adjacency between vertices of ψ(G 1, G 2, G 3) is defined by: 1. (u i , w l ) adj (v k , w l ) for each i = 1, 2,..., n 1 and k = 1, 2,..., n 2 and l = 1, 2,..., n 3. 2. (v i , w j ) adj (v k , w l ) if and only if i) i = k and w j adj w l , in G 3 or ii) j = l and v i adj v k in G 2 3. (u i , w j ) adj (u k , w j ) whenever u i adj u k in G 1, for each j = 1, 2,..., n 3. We obtain the Adjacency spectrum, Laplacian spectrum and Q-spectrum of ψ(G 1, G 2, G 3). As an application, many new infinite families of R-integral graphs where R ∈ {A, L, Q} are constructed.