Unit graphs of rings of polynomials and power series

Abstract

Let R be a commutative ring. The unit graph of R, denoted by G(R), is a graph with all elements of R as vertices and two distinct vertices x, y ∈ R are adjacent if and only if x + y ∈ U(R) where U(R) denotes the set of all units of R. In this paper, we examine the preservation of the connectedness, diameter, girth, and some other properties, such as chromatic index, clique number and planarity of the unit graph G(R) under extensions to polynomial and power series rings. [Figure not available: see fulltext.]