Some applications of eigenvalues of unitary Cayley graphs of matrix rings over finite fields (Q6638364)

The unitary Cayley graph of a finite ring \(R\) is the graph whose vertex set is \(R\) and for each \(a, b \in R\), \(a\) is adjacent to \(b\) if and only if \(a-b\) is a unit of \(R\). A ring \(R\) is a DU-ring if for any ring \(S\) such that \(R\) and \(S\) have isomorphic unitary Cayley graphs, \(R\) and \(S\) are isomorphic themselves. By examining the eigenvalues of the matrix ring over a field, the authors in this paper obtain many new families of DU-rings.\N\NWe say that a \(k\)-regular graph \(G\) is Ramanujan if \(|\lambda| \leq 2\sqrt{k-1}\) for all eigenvalues \(\lambda\) of \(G\) other than \(\pm k\). The authors in this paper characterize direct products of matrix rings over local rings such that their unitary Cayley graph is Ramanujan.