The matrix Jacobson graph of finite commutative rings

Abstract

The notion of the matrix Jacobson graph was introduced in 2019. Let R be a commutative ring and J ( R ) be the Jacobson radical of ring R . The matrix Jacobson graph of ring R size m × n , denoted 𝔍( R ) m × n , is defined as a graph where the vertex set is R m × n ∖ J ( R ) m × n such that two distinct vertices A , B are adjacent if and only if 1 − det( A t B ) is not a unit in ring R . Here we obtain some graph theoretical properties of 𝔍( R ) m × n including its connectivity, planarity and perfectness.