The Size of Maximally Irregular Graphs and Maximally Irregular Triangle-Free Graphs

Let G be a graph. The irregularity index of G, denoted by t(G), is the number of distinct values in the degree sequence of G. For any graph G, t(G) ≤ Δ(G), where Δ(G) is the maximum degree. If t(G) = Δ(G), then G is called maximally irregular. In this paper, we give a tight upper bound on the size of maximally irregular graphs, and prove the conjecture proposed in [6] on the size of maximally irregular triangle-free graphs. Extremal graphs are also characterized.