When the maximal graph is planar, outerplanar, and ring graph

Abstract

Let [Formula: see text] be a commutative ring with nonzero identity. Let [Formula: see text] denote the maximal graph associated to [Formula: see text], that is, [Formula: see text] is a graph with vertices as non-units of [Formula: see text], where two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if there is a maximal ideal of [Formula: see text] containing both. In this paper, we characterize the finite commutative rings such that their maximal graph are planar graphs, and we also study the case where they are outerplanar and ring graphs. The equivalence of outerplanar graphs and ring graphs for [Formula: see text] is established.