In this paper we investigate geometric properties of graphs generated by a preferential attachment random graph model with edge-steps. More precisely, at each time $t\in\mathbb{N}$, with probability $p$ a new vertex is added to the graph (a vertex-step occurs) or with probability $1-p$ an edge connecting two existent vertices is added (an edge-step occurs). We prove that the global clustering coefficient decays as $t^{-γ(p)}$ for a positive function $γ$ of $p$. We also prove that the clique number of these graphs is, up to sub-polynomially small factors, of order~$t^{(1-p)/(2-p)}$.