Irregularity and Near Star Graphs

where the sum is over all edges of G.In 2005, Hansen and Mélot proved an upper bound on the irregularity of a graph on n vertices and m edges that is attained for all feasible values, and characterized the extremal graphs.We give a much shorter proof of this result, and discuss some of its implications.Denoting the extremal graphs as near k-star graphs, we show that they sometimes occur as extremal graphs for related problems on sigma irregularity and first Zagreb index.