The trace graph of the matrix ring over a finite commutative ring (Q1714957)

Let \(R\) be a commutative ring, and let \(n\geq 2\) be an integer. The authors consider a graph whose vertices are all non-zero matrices \(A\in M_n(R)\) such that there exists a non-zero \(B\in M_n(R)\) with \(\mathrm{tr}(AB)=0\). Two vertices \(A\) and \(B\) are joined by an edge whenever \(\mathrm{tr}(AB)=0\). It is proved that this graph is connected, and it has diameter 2 and girth 3. In the case where \(R\) is a finite ring, it is described when the irregularity index of this graph is 2 or 3.