Finite Euclidean graphs over rings

Abstract

Abstract. We consider graphs attached to (Z/qZ) n , where q = p r , for an odd prime p, using an analogue of the Euclidean distance. These graphs are shown to be mostly non-Ramanujan, in contrast to the case of Euclidean graphs over finite fields.

Eigenvalues of Cayley graphs

“…Similar to E q (n, a), for an odd prime power q and a positive integer n, the Euclidean graph associated with Z n q and a ∈ Z q is defined [236] to be the Cayley graph…”

“…In [236], Medrano et al determined the spectra and Ramanujancy of X q (n, a). See also [100] and [298].…”

On the spectrum of unitary finite-Euclidean graphs

“…In [12], Medrano, Myers, Stark and Terras studied the spectrum of the finite-Euclidean graphs over finite fields and showed that these graphs are asymptotically Ramanujan graphs. In [13], these authors studied the same problem for the finite-Euclidean graphs over rings q for q odd prime power. They showed that, over rings, except for the smallest case, the graphs (with unit distance parameter) are not (asymptotically) Ramanujan.…”