Some results on the Jacobson graph of a commutative ring

Abstract

Let R be a commutative ring with non-zero identity and J(R) be Jacobson ideal of R. The Jacobson graph of R is the graph whose vertices are \(R{\setminus } J(R)\), and two different vertices x and y are adjacent if \(1-xy\notin U(R)\), where U(R) is the set of units of R. We investigate diameter of \(\mathfrak {J}_R\) and seek relation between it and diameter of Jacobson graphs under extension to polynomial and power series rings. Also, vertex and edge connectivity of finite Jacobson graphs are obtained. Finally, we show that all finite Jacobson graphs have a matching that misses at most one vertex and offer one 1-factor decomposition of a regular induced subgraph.