Ideal based trace graph of matrices

Abstract

Let $R$ be a commutative ring and $M_n(R)$ be the set of all $n\times n$ matrices over $R$ where $n\geq 2.$ The trace graph of the matrix ring $M_n(R)$ with respect to an ideal $I$ of $R,$ denoted by $\Gamma_{I^t}(M_n(R)),$ is the simple undirected graph with vertex set $M_n(R)\setminus M_n(I)$ and two distinct vertices $A$ and $B$ are adjacent if and only if Tr$(AB) \in I.$ Here Tr$(A)$ represents the trace of the matrix $A.$ In this paper, we exhibit some properties and structure of $\Gamma_{I^t}(M_n(R)).$