Extremal values for the maximum degree in a twin-free graph

Consider a connected undirected graph G = (V, E) and an integer r ≥ 1; for any vertex v ε V, let B_r(v) denote the ball of radius r centred at v, i.e., the set of all vertices linked to v by a path of at most r edges. If for all vertices v ε V, the sets B_r(v) are different, then we say that G is r-twin-free. In r-twin-free graphs, we prolong the study of the extremal values that can be reached by some classical parameters in graph theory, and investigate here the maximum degree.