Mean eccentricities of de Bruijn networks

Given a graph G = (V,E) we define e(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 e(X). For the directed de Bruijn network, we provide tight bounds as well as the extremal vertices which reach these bounds. The 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 a . . . a have the largest mean eccentricity. Numerical computations indicate that the conjecture holds for binary de Bruijn networks with diameters up to 18. We prove that the asymptotic difference, when the diameter goes to infinity, between the mean eccentricities of an arbitrary vertex and that of a . . . a is smaller than a small constant tending to zero with the degree. We also provide a simple recursive sheme for the computation of the asymptotic mean eccentricity of the vertices a . . . a. A by-product 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 tend to the diameter when the degree goes to infinity