Summary: J. Bond [Grands réseaux d’interconnexion. Paris: Paris-sud University (PhD thesis) (1987)] and J. Bond et al. [The radius of graphs on alphabets. Rapport de Recherche 156-RR 388, Paris-sud University], conjectured that a quasi-center in an undirected de Bruijn graph \(\mathrm{UB}(d,D)\) has cardinality at least \(d-1\), and that a quasi-center in an undirected Kautz graph \(\mathrm{UK}(d,D)\) has cardinality at least \(d\). They proved that for \(d\geq 3\), the radii of \(\mathrm{UB}(d,D)\) and \(\mathrm{UK}(d,D)\) are both equals to \(D\), and conjectured also that the radii of \(\mathrm{UB}(2,D)\) and \(\mathrm{UK}(2,D)\) are respectively \(D-1\) and \(D\). In this paper we give results in a more general context which validate these conjectures (excepting that asserting that the radius of \(\mathrm{UB}(2,D)\) is \(D-1\)), and give simplified proofs of the cited results.