ON RELATION BETWEEN ENERGY AND LAPLACIAN ENERGY

Let G be a simple graph with n vertices and m edges, with ordinary spectrum lambda(i), i = 1, 2, ... , n., and with Laplacian spectrum mu(i), i = 1, 2, ... , n. The energy and the Laplacian energy of the graph G are defined as E(G) = Sigma(n)(i=1) vertical bar lambda(i)vertical bar and LE(G) = Sigma(n)(i=1) vertical bar mu(i) - 2m/n vertical bar, respectively. In [9] the authors provided numerous examples for the inequality E(G) <= LE(G) and conjectured that it holds for all graphs. In this paper we show that the conjecture does not hold.