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

## TL;DR

This paper determines the eigenvalues of the graphs D(4,q), revealing they are excellent expanders close to Ramanujan graphs, which enhances understanding of their spectral and expansion properties.

## Contribution

The paper computes the spectrum of D(4,q) graphs, the smallest open case, showing they are nearly Ramanujan expanders with specific eigenvalue bounds.

## Key findings

- Eigenvalues other than ±q are bounded by 2√q
- Graphs are q-regular on 2q^4 vertices
- Graphs are nearly Ramanujan expanders

## Abstract

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.