Spectra of unitary Cayley graphs of matrices over finite commutative rings

Abstract

For a finite ring R with identity, the unitary Cayley graph of R, CR, is the graph with vertex set R and for each x, y ∈ R, x and y are adjacent if and only if x − y is a unit of R. In this thesis, we determine some eigenvalues of CMn(F) , where F is a finite field, by using the additive characters and use these eigenvalues to analyze strong regularity, hyperenergetic graphs and Ramanujan graphs. Next, we extend the results to CMn(R) , where R is a local ring. We characterize all local rings R and n ≥ 2 such that the graph CMn(F) is strongly regular and Ramanujan and also show that the graph is hyperenergetic. Moreover, we show that the unitary Cayley graph of product of matrix rings is hyperenergetic. Finally, we prove that the unitary Cayley graph of product of matrix rings is neither a strongly regular graph nor a Ramanujan graph