Some new spectral bounds for graph irregularity

The irregularity of a simple graph G=(V,E) is defined as irr(G)=∑uv∈E(G)|dG(u)−dG(v)|,where dG(u) denotes the degree of a vertex u ∈ V(G). This graph invariant, introduced by Albertson in 1997, is a measure of the defect of regularity of a graph. Recently, it also gains interest in Chemical Graph Theory, where it is named the third Zagreb index. In this paper, by means of the Laplacian eigenvalues and the normalized Laplacian eigenvalues of G, we establish some new spectral upper bounds for irr(G). We then compare these new bounds with a known bound by Goldberg, and it turns out that our bounds are better than the Goldberg bound in most cases. We also present two spectral lower bounds on irr(G).