A note on the LEL-equienergetic graphs

Let G be a graph with n vertices and μ1, μ2, ..., μn be the Laplacian eigenvalues of G. The Laplacian-energy-like graph invariant LEL(G) = ∑n i=1 √ μi, has been defined and investigated in [1]. Two non-isomorphic graphs G1 and G2 of the same order are said to be LEL-equienergetic if LEL(G1) = LEL(G2). In [2], three pairs of LELequienergetic non-cospectral connected graphs are given. It is also claimed that the LEL-equienergetic non-cospectral connected graphs are relatively rare. It is natural to consider the question: Whether the number of the LEL-equienergetic non-cospectral connected graphs is finite? The answer is negative, because we shall construct a pair of LEL-equienergetic non-cospectral connected graphs of order n, for all n ≥ 12 in this paper.