Counting irregular multigraphs

Gagliardi et al. (1996, unpublished manuscript) defined an irregular multigraph to be a loopless multigraph with degree sequence n, n − 1,…, 1, and they posed the problem of determining the number of different irregular multigraphs fn on n vertices. In Gagliardi et al. (1996) they showed that if n ≡ 0 or 3 (mod 4) then fn > n − 1. In this note our aim is to show that there are constants 1 < c1 < c2 and n0 > 0 such that if n ⩾ n0 and n ≡ 0 or 3 (mod 4) then (c1)n2 < fn < (c2)n2. Indeed, we show that c1 = 1.19 and c2 = 1.65 can be chosen.