Abstract. For a simple connected graph G of order n, having Laplacian eigenvalues µ 1 , µ 2 , . . . , µ n−1 , µn = 0, the Laplacian-energy-like invariant (LEL) and the Kirchhoff index (Kf ) are defined as, respectively. In this paper, LEL and Kf are compared, and sufficient conditions for the inequality Kf (G) < LEL(G) are established.

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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 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 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.

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