Laplacian energy of union and cartesian product and laplacian equienergetic graphs

Abstract

The Laplacian energy of a graph G with n vertices and m edges is defined as LE(G) = ∑ni=1 |μi-2m/n|, where μ1, μ2,...,μn are the Laplacian eigenvalues of G. If two graphs G1 and G2 have equal average vertex degrees, then LE(G1 ∪ G2) = LE(G1) + LE(G2). Otherwise, this identity is violated. We determine a term Ξ, such that LE(G1) + LE(G2) - Ξ ≤LE(G1 ∪ G2) ≤ LE(G1)+LE(G2)+Ξ holds for all graphs. Further, by calculating LE of the Cartesian product of some graphs, we construct new classes of Laplacian non-cospectral, Laplacian equienergetic graphs.