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 + alpha(mod m)\alpha is an element of [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 will show that the diameter of UG(B) (n, m) is 2 for any m in [n + 1, n(2)] where n divides m and that the diameter is 3 for any m in [n(2) + 1, n] where n divides m.