On the spectrum of the closed unit graphs

Abstract

Let R be a finite commutative ring with non-zero identity. Let $R^{\times }$ and $J\left(R\right)$ be the group of unit elements and the Jacobson radical of R, respectively. The unit graph of the ring R, denoted by $G\left(R\right)$, is a graph whose vertex set is R and two distinct vertices x and y are adjacent if and only if $x+y\in R^{\times }$. If we relax this definition by dropping the term ‘distinct’, we obtain the closed unit graph, denoted by $\overline{G}\left(R\right)$. In this paper, we compute the adjacency spectrum of the graph $\overline{G}\left(R\right)$. We utilize this result to show that $G\left(R\right)\cong G\left(S\right)$ if and only if $\left(R/J\left(R\right)\right)\cong \left(S/J\left(S\right)\right)$ and $|J\left(R\right)|=|J\left(S\right)|$, where R and S are two arbitrary finite rings. Moreover, we determine when $G\left(R\right)$ is a Ramanujan graph. We also deliver a necessary and sufficient condition for $G\left(R\right)$ to be a strongly regular graph. Finally, we obtain the spectrum of a generalization of both unit and unitary Cayley graphs.

Keywords:

• Unit graphs
• spectrum
• isomorphism
• strongly regular graphs
• finite rings

2010 Mathematics Subject Classifications:

• 05C25
• 05C40
• 05C75
• 13M05
• 16U60

Disclosure statement

No potential conflict of interest was reported by the author(s).