On maximally irregular graphs

Let G be a connected graph with maximum degree ?(G). The irregularity index t(G) of G is defined as the number of distinct terms in the degree sequence of G. We say that G is maximally irregular if t(G) = ?(G). The purpose of this note, apart from pointing out that every highly irregular graph is maximally irregular, is to establish upper bounds on the size of maximally irregular graphs and maximally irregular triangle-free graphs.