Abstract
Let R be a finite commutative ring with non-zero identity. Let and
be the group of unit elements and the Jacobson radical of R, respectively. The unit graph of the ring R, denoted by
, is a graph whose vertex set is R and two distinct vertices x and y are adjacent if and only if
. If we relax this definition by dropping the term ‘distinct’, we obtain the closed unit graph, denoted by
. In this paper, we compute the adjacency spectrum of the graph
. We utilize this result to show that
if and only if
and
, where R and S are two arbitrary finite rings. Moreover, we determine when
is a Ramanujan graph. We also deliver a necessary and sufficient condition for
to be a strongly regular graph. Finally, we obtain the spectrum of a generalization of both unit and unitary Cayley graphs.
Disclosure statement
No potential conflict of interest was reported by the author(s).