Bounds and Computation of Irregularity of a Graph

Albertson [3] has defined the irregularity of a simple undirected graph G = (V,E) as irr(G) = ∑ uv∈E |dG(u)− dG(v)| , where dG(u) denotes the degree of a vertex u ∈ V . Recently, this graph invariant gained interest in the chemical graph theory, where it occured in some bounds on the first and the second Zagreb index, and was named the third Zagreb index [13]. For general graphs with n vertices, Albertson has obtained an asymptotically tight upper bound on the irregularity of 4n/27. Here, by exploiting a different approach than in [3], we show that for general graphs with n vertices the upper bound bn3 cd 2n 3 e ( d 2n 3 e − 1 ) is sharp. Next, we determine k-cyclic graphs with maximal irregularity. We also present some bounds on the maximal/minimal irregularity of graphs with fixed minimal and/or maximal vertex degrees, and consider an approximate computation of the irregularity of a graph.