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Abstract. In this article we generalize the concept of line digraphs to line dihypergraphs. We give some general properties in particular concerning connectivity parameters of dihypergraphs and their line dihypergraphs, like the fact that the arc connectivity of a line dihypergraph is greater than or equal to that of the original dihypergraph. Then we show that the De Bruijn and Kautz dihypergraphs (which are among the best known bus networks) are iterated line digraphs. Finally we give short proofs that they are highly connected.

Given a graph G Å (V, E), we define eV ( X ), the mean eccentricity of a vertex X, as the average distance from X to all the other vertices of the graph. The computation of this parameter appears to be nontrivial in the case of the de Bruijn networks. In this paper, we consider upper and lower bounds for eV ( X ). For the directed de Bruijn network, we provide tight bounds as well as the extremal vertices which reach these bounds. These bounds are expressed as the diameter minus some constants. In the case of undirected networks, the computation turns out to be more difficult. We provide lower and upper bounds which differ from the diameter by some small constants. We conjecture that the vertices of the form arrra have the largest mean eccentricity. Numerical computations indicate that the conjecture holds for binary de Bruijn networks with diameters up to 18. We also provide a simple recursive scheme for the computation of the asymptotic mean eccentricity of the vertices arrra. Finally, we prove that the asymptotic difference, when the diameter goes to infinity, between the mean eccentricities of an arbitrary vertex and that of arrra is smaller than a small constant tending to zero with the degree. A byproduct of our analysis is that in both directed and undirected de Bruijn networks most of the vertices are at distance near from the diameter and that all of the mean eccentricities (and therefore the average distance) tend to the diameter when the degree goes to infinity.