A Note on the LEL-Equienergetic Graphs

Abstract

Let \(G\) be a graph with \(n\) vertices and \(\mu_1, \mu_2, \ldots, \mu_n\) be the Laplacian eigenvalues of \(G\). The Laplacian-energy-like graph invariant \(\text{LEL}(G) = \sum_{i=1}^{n} \sqrt{\mu_i}\) has been defined and investigated in [1]. Two non-isomorphic graphs \(G_1\) and \(G_2\) of the same order are said to be \(\text{LEL}\)-equienergetic if \(\text{LEL}(G_1) = \text{LEL}(G_2)\). In [2], three pairs of \(\text{LEL}\)-equienergetic non-cospectral connected graphs are given. It is also claimed that the \(\text{LEL}\)-equienergetic non-cospectral connected graphs are relatively rare. It is natural to consider the question: Whether the number of the \(\text{LEL}\)-equienergetic non-cospectral connected graphs is finite? The answer is negative, because we shall construct a pair of \(\text{LEL}\)-equienergetic non-cospectral connected graphs of order \(n\), for all \(n \geq 12\) in this paper.