The Eigenvalues of the Graphs $D(4,q)$

The graphs $D(k,q)$ have connected components $CD(k,q)$ giving the best known bounds on extremal problems with {\em forbidden\/} even cycles, and are denser than the well-known graphs of Lubotzky, Phillips, Sarnak and Margulis. Despite this, little about the spectrum and expansion properties of these graphs is known. In this paper we find the spectrum for $k=4$, the smallest open case. For each prime power $q$, the graph $D(4,q)$ is $q$-regular graph on $2q^4$ vertices, all of whose eigenvalues other than $\pm q$ are bounded in absolute value by $2\sqrt{q}$. Accordingly, these graphs are good expanders, in fact very close to Ramanujan.