In 1995, Lazebnik and Ustimenko introduced the family of $q$-regular graphs $D(k, q)$, which is defined for any positive integer $k$ and prime power $q$. The connected components of the graph $D(k, q)$ have provided the best-known general lower bound on the size of a graph for any given order and girth to this day. Furthermore, Ustimenko conjectured that the second largest eigenvalue of $D(k, q)$ is always less than or equal to $2\sqrt{q}$, indicating that the graphs $D(k, q)$ are almost Ramanujan graphs. In this talk, we will discuss some recent progress on this conjecture. This includes the result that the second largest eigenvalue of $D(5, q)$ is less than or equal to $2\sqrt{q}$ when $q$ is an odd prime power. This is joint work with Vladislav Taranchuk.