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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is the distance matrix and Tr(G) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α (G). Some new upper and lower bounds for the generalized distance energy E D α (G) of G are established based on parameters including the Wiener index W(G) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

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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