A note on unitary Cayley graphs of matrix algebras

TL;DR

This paper proves that the unitary Cayley graph of matrix algebras over finite fields is strongly regular exclusively when the matrix size is 2, clarifying previous partial results and filling a gap in the literature.

Contribution

It establishes a complete characterization of when the unitary Cayley graph of matrix algebras over finite fields is strongly regular, extending prior partial findings.

Findings

Strongly regular only for n=2

General case for n>2 is not strongly regular

Fills gap in the classification of these graphs

Abstract

Dariush Kiani et al.\cite{kiani2015unitary} claim to have found the unitary Cayley graph $Cay(M_{n}(F),GL_{n}(F))$ of matrix algebras over finite field $F$ is strongly regular only when $n=2$, but they have only considered two special cases, namely when $n = 2$ and $3$ and have failed to cover the general cases. In this paper, we prove that the unitary Cayley graph of matrix algebras over finite field $F$ is strongly regular if and only if $n=2$.