On the Spectra of Certain Graphs Arising from Finite Fields

Cayley graphs on a subgroup ofGL(3,p),p>3 a prime, are defined and their properties, particularly their spectra, studied. It is shown that these graphs are connected, vertex-transitive, nonbipartite, and regular, and their degrees are computed. The eigenvalues of the corresponding adjacency matrices depend on the representations of the group of vertices. The “1-dimensional” eigenvalues can be completely described, while a portion of the “higher dimensional” eigenfunctions are discrete analogs of Bessel functions. A particular subset of these graphs is conjectured to be Ramanujan and this is verified for over 2000 graphs. These graphs follow a construction used by Terras on a subgroup ofGL(2,p). This method can be extended further to construct graphs using a subgroup ofGL(n, p) forn≥4. The 1-dimensional eigenvalues in this case can be expressed in terms of the 1-dimensional eigenvalues of graphs fromGL(2,p) andGL(3,p); this part of the spectra alone is sufficient to show that forn≥4, the graphs fromGL(n, p) are not in general Ramanujan.