A spectral bound for graph irregularity

The imbalance of an edge e = { u , v } in a graph is defined as i ( e ) = | d ( u )− d ( v )|, where d (·) is the vertex degree. The irregularity I ( G ) of G is then defined as the sum of imbalances over all edges of G . This concept was introduced by Albertson who proved that I ( G ) ⩽ 4 n 3 /27 (where n = | V ( G )|) and obtained stronger bounds for bipartite and triangle-free graphs. Since then a number of additional bounds were given by various authors. In this paper we prove a new upper bound, which improves a bound found by Zhou and Luo in 2008. Our bound involves the Laplacian spectral radius λ.