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

ABSTRACT

In this work, we provide two applications of the eigenvalues of the unitary Cayley graphs over matrix rings over finite fields. For a ring R, $J_R$ denotes the Jacobson radical of R. In 2012, Kiani and Aghaei conjectured that if R and S are finite rings and their unitary Cayley graphs are isomorphic, then $R/J_R$ and $S/J_S$ are isomorphic. If this conjecture holds and $J_R=\left\{0\right\}$, then R is characterized by its unitary Cayley graph, and we say that R is a ring determined by unitary Cayley graphs. Kiani and Aghaei showed that every finite commutative ring and ${\text{M}}_n\left({\mathbb{F}}_q\right)$ are such rings. We examine the eigenvalues of ${\text{M}}_n\left({\mathbb{F}}_q\right)$ and obtain many new families of rings determined by unitary Cayley graphs. In 2021, Podestá and Videla characterized all finite commutative rings R such that the graphs in the triple $\{{\text{C}}_R,{\text{C}}_R^{+},{\overline{\text{C}}}_R\}$ are mutually equienergetic non-isospectral and Ramanujan where ${\text{C}}_R^{+}$ is the unitary Cayley sum graph. We use the eigenvalues of the graph ${\text{C}}_{{\text{M}}_n\left({\mathbb{F}}_q\right)}$ and some observation on the number of matrices of the given rank to extend Podestá and Videla's results by working on the finite non-commutative ring $\mathcal{R}$ being a product of matrix rings over local rings.

KEYWORDS:

• DU-ring
• semisimple ring
• Ramanujan graph
• unitary Cayley graph

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 05C50
• 05C25

Acknowledgments

The authors thank anonymous referees for their valuable comments.

Disclosure statement

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

Additional information

Funding

The first author is supported in part by the Research Assistantship Funding (RAF_2565_01) from Faculty of Science, Chulalongkorn University.