Supersingular elliptic curve isogeny graphs have isomorphism classes of supersingular elliptic curves over a finite field as their vertices and isogenies of some fixed degree between them as their edges. Due to their apparent "random" nature, supersingular isogeny graphs - which are optimal expander graphs - have been used as a setting for certain cryptographic schemes that are resistant to attacks by quantum computers. Hidden structures in these graphs may have implications to the security of these systems. In this talk, we analyze a number of graph theoretic structural properties of supersingular isogeny graphs over a finite field $\mathbb{F}_{p^2}$ and their subgraphs induced by the vertices defined over $\mathbb{F}_p$. This is joint work with Sarah Arpin (Virginia Tech) and our jointly supervised undergraduate student Taha Hedayat (University of Calgary).