Large sets of noncospectral graphs with equal laplacian energy

Abstract

Several alternative definitions to graph energy have appeared in literature recently, the first among them being the Laplacian energy, defined by Gutman and Zhou in [Linear Algebra Appl. 414 (2006), 29-37]. We show here that Laplacian energy apparently has small power of discrimination among threshold graphs, by showing that, for each n, there exists a set of n mutually noncospectral connected threshold graphs with equal Laplacian energy with O(√n) vertices only. Nevertheless, situation becomes opposite when trees are considered, as it turns out that, up to 20 vertices, there exists no pair of noncospectral trees with equal Laplacian energies.