Some Lower Bounds for Laplacian Energy of Graphs

We consider only finite undirected graphs without loops or multiple edges. Notation and terminology not defined here follow that in [1]. Let G be a graph of order n and size m. We assume that d1, d2, ..., dn, where di, 1 ≤ i ≤ n, is the degree of vertex vi in G, is the degree sequence of G. We define Σk(G) as ∑n i=1 d k i . For each vertex vi, 1 ≤ i ≤ n, mi is defined as the sum of degrees of vertices that are adjacent to vi. Obviously, Σ n i=1mi = Σ n i=1d 2 i = Σ2(G). The Laplacian of a graph G is defined as L(G) = D(G)−A(G), where D(G) is the diagonal matrix of the degree sequence of G and A(G) is the adjacency matrix of G. The eigenvalues λ1(G) ≥ λ2(G), ...,≥ λn(G) = 0 of L(G) are called the Laplacian eigenvalues of the graph G. The Laplacian energy of a graph G is defined as LE(G) = ∑n i=1 |λi − 2m n |. This concept was introduced by Gutman and Zhou in [4] and more detailed discussions on LE(G) can be found in [4].