On the diameter of the generalized undirected de Bruijn graphs UGB(n, m), n2 < m <= n3

The generalized deBruijn digraphGB(n,m) is the digraph (V ,A)where V = {0, 1, . . . ,m−1} and (i , j ) ∈ A if and only if j ≡ in + α (mod m) for some α ∈ {0, 1, 2, . . . ,n − 1}. By replacing each arc of GB(n,m) with an undirected edge and eliminating loops and multi-edges, we obtain the generalized undirected de Bruijn graph UGB(n,m). In this article, we prove that when 2n2 ≤ m ≤ n3 the diameter of UGB(n,m) is equal to 3. We also show that for pairs (n,m)where n2 < m < 2n2 the diameter ofUGB(n,m) can be 2 or 3. © 2008 Wiley Periodicals, Inc. NETWORKS, Vol. 52(4), 180–182 2008