Moments of graphs in monotone families

The k th moment of the degree sequence d 1  ≥  d 2  ≥ … d n of a graph G is $\mu _k(G)={1\over n}{\sum}{d_i^k}$ . We give asymptotically sharp bounds for μ k ( G ) when G is in a monotone family. We use these results for the case k  = 2 to improve a result of Pach, Spencer, and Tóth [15]. We answer a question of Erdős [9] by determining the maximum variance ${\mu _2(G)-\mu _1^2(G)}$ of the degree sequence when G is a triangle‐free n ‐vertex graph. © 2005 Wiley Periodicals, Inc.