Edge number, minimum degree, maximum independent set, radius and diameter in twin-free graphs

Consider a connected, undirected graph G = (V,E) and an integer r ≥ 1; for any vertex ν ∈ V, let B_r(ν) denote the ball of radius r centred at ν, i.e., the set of all vertices linked to ν by a path consisting of at most r edges. If for all vertices ν ∈ V, the sets B_r(ν) 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 achieved by the main classical parameters in graph theory, and investigate here the number of edges, the minimum degree, the size of a maximum independent set, as well as radius and diameter.