Graph-theoretical properties of supersingular isogeny graphs of elliptic curves

In this thesis we present isogeny graphs of elliptic curves, after introducing the necessary preliminary notions in graph theory and elliptic curves. We focus on supersingular isogeny graphs over $\mathbb{F}_p$ and over $\overline{\mathbb{F}}_p$ and we describe some of their properties: regularity, expansion, undirectedness. We carefully explain why a supersingular isogeny graph over $\overline{\mathbb{F}}_p$ can be attributed the Ramanujan property, even though this is in general a directed graph. We finally explain how defining edges as isogenies up to pre and post-composition with automorphisms, rather than just post-composition, affects the features of supersingular isogeny graphs.