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, . . . , xn} and Y = {y1, y2, . . . , yn}, where two vertices xi and yj are adjacent if and only if i = j or vi adjacent to vj in G. The double graphD[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 Gk∗ 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.