Energy, Laplacian energy of double graphs and new families of equienergetic graphs

For a graph G with vertex set V(G) = {v1, v2, . . . , vn}, the extended double cover G* is a bipartite graph with bipartition (X, Y), X = {x₁, x₂, . . . , x_n} and Y = {y₁, y₂, . . . , y_n}, where two vertices x_i and y_j are adjacent if and only if i = j or vi adjacent to v_j in G. The double graph D[G] of G is a graph obtained by taking two copies of G and joining each vertex in one copy with the neighbors of corresponding vertex in another copy. In this paper we study energy and Laplacian energy of the graphs G* and D[G], L-spectra of G^k* the k-th iterated extended double cover of G. We obtain a formula for the number of spanning trees of G*. We also obtain some new families of equienergetic and L-equienergetic graphs.