In this talk we will be interested in describing the distribution of the maximal degree of vertices in a large random graph. More specifically, we will be interested in the Poisson-Delaunay and the Poisson-Beta(-prime)-Delaunay graphs. The constructions of these graphs are based on an underlying Poisson point process in the Euclidean space and following specific geometric rules to construct the set of edges. We want to describe the distribution of the maximal degree of all the vertices within a large observation window. We will first present results obtained with Nicolas Chenavier (Bernoulli 2020) showing a concentration on finitely many values in the Poisson-Delaunay model, as the window?s size goes to infinity. Then, we will discuss the ongoing work with my PhD student (Joseph Gordon) in the case of the Poisson-Beta-Delaunay and Poisson-Beta-prime-Delaunay models. Joint work with Nicolas Chenavier and Joseph Gordon