MINIMAL LEL-EQUIENERGETIC GRAPHS

Let G be a graph with n vertices and m edges. Let μ1 ,μ 2 ,...,μ n be the Laplacian eigenvalues of G . The Laplacian–energy–like graph invariant LEL(G )= " n=1 √ μi , has been recently defined and investigated by two of present authors [J. Liu, B. Liu, MATCH Commun. Math. Comput. Chem. 59 (2008) 355–372.]. Two non-isomorphic graphs G1 and G2 of the same order are said to be LEL-equienergetic if LEL(G1 )= LEL(G2) . In this paper we consider LEL-equienergetic and almost–LEL-equienergetic graphs and find that they occur relatively rarely. If the criterion for almost–LEL-equienergeticity is |LEL(G1) − LEL(G2)| < 10 −8 , then for n ≤ 14 there are no pairs of noncospectral LEL-equienergetic or almost–LEL-equienergetic trees, but there exist two almost–equienergetic pairs for n =1 5 .

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