On the diameter of a class of the generalized de Bruijn graphs

The generalized de Bruijn digraph denoted by G_B(n, m) is defined to be the digraph with m vertices labelled by 0, 1, 2, ..., m-1 and with the adjacency defined as follows: If i is a vertex in G_B(n, m) then i is connected to each vertex in the set E(i), where E(i)={ni+α(mod m)|α∈[0, n-1]}. The generalized de Bruijn graph denoted by UG_B(n, m) is defined to be the undirected version of G_B(n, m) obtained by replacing each arc by an undirected edge and eliminating self-loops and multi-edges. In this paper we show that the diameter of UG_B(n, m) is 2 for any m in [n+1, n²] where n divides m and that the diameter is 3 for any m in [n²+1, n³] where n divides m