Let G be a graph with n vertices and m edges. Let λ1, λ2, . . . , λn be the eigenvalues of the adjacency matrix of G, and let μ1, μ2, . . . , μn be the eigenvalues of the Laplacian matrix of G. An earlier much studied quantity E(G) = ∑ni=1 |λi | is the energy of the graph G. We now define and investigate the Laplacian energy as LE(G) = ∑ni=1 |μi − 2m/n|. There is a great deal of analogy between...

In this paper, we investigate how the Laplacian coefficients changed after some graph transformations. So, I express some results about Laplacian coefficients ordering of graphs, focusing our attention to the bicyclic graphs. Finally, as an application of these results, we discuss the ordering of graphs based on their Laplacian like energy.

A b s t r a c t. Let G denote the complement of the graph G . If I(G) is some invariant of G , then relations (identities, bounds, and similar) pertaining to I(G) + I(G) are said to be of Nordhaus-Gaddum type. A number of lower and upper bounds of Nordhaus-Gaddum type are obtained for the energy and Laplacian energy of graphs. Also some new relations for the Laplacian graph energy are established.

Let G be a simple connected graph with n ≤ 2 vertices and m edges, and let μ1 ≥ μ2 ≥...≥μn-1 >μn=0 be its Laplacian eigenvalues. The Kirchhoff index and Laplacian-energy-like invariant (LEL) of graph G are defined as Kf(G)=nΣi=1n-1<...