Quasi-centers and radius related to some iterated line digraphs, proofs of several conjectures on de Bruijn and Kautz graphs

Bond (1987) and Bond et al. (1987), conjectured that a quasi-center in an undirected de Bruijn graph UB(d,D)UB(d,D) has cardinality at least d−1d−1, and that a quasi-center in an undirected Kautz graph UK(d,D)UK(d,D) has cardinality at least dd. They proved that for d≥3d≥3, the radii of UB(d,D)UB(d,D) and UK(d,D)UK(d,D) are both equals to DD, and conjectured also that the radii of UB(2,D)UB(2,D) and UK(2,D)UK(2,D) are respectively D−1D−1 and DD. In this paper we give results in a more general context which validate these conjectures (excepting that asserting that the radius of UB(2,D)UB(2,D) is D−1D−1), and give simplified proofs of the cited results.