Finite commutative rings with higher genus unit graphs

Abstract

Communicated by X. D. HouLet R be a ring with identity. The unit graph of R, denoted by G(R), is a simple graph with vertex set R, and where two distinct vertices x and y are adjacent if and only if x + y is a unit in R. The genus of a simple graph G is the smallest nonnegative integer g such that G can be embedded into an orientable surface Sg . In this paper, we determine all isomorphism classes of finite commutative rings whose unit graphs have genus at most three.

Graphs Defined on Rings: A Review

“…In succession to the planar unit graphs, the non-planar unit graphs of finite commutative rings that had genus one were investigated in [243], where all finite commutative rings with non-zero identity whose unit graphs were toroidal were determined up to isomorphism, and it was proven that, for any positive integer k, there are finitely many finite commutative rings with non-zero identity, such that the genus of their unit graph is k. As a continuation of the study on the unit graphs of finite commutative rings with unit genus, the rings having unit graphs with higher order genera were investigated in [244], and all finite rings with unit graphs having genera one, two, and three were characterised.…”

A Note on the Double Total Graph T_u(Γ(R)) and T_u(Γ(Z_n×Z_m))

“…Let R be a finite commutative ring with unity. By [7,Theorem 2.11(2)], [2,Theorem 5.14] and [6,Proposition 4.1], it is enough to consider the following rings as G(R) is a spanning subgraph of T u (Γ(R)).…”

The Diameter of Unit Graphs of Rings

“…Afkhami and Khosh-Ahang studied the unit graphs of rings of polynomials and power series in [1]. In 2014, Su and Zhou [18] proved that the girth of G(R) is 3, 4, 6 or ∞ for an arbitrary ring R. Other papers are also devoted to this topic (see, [2,16,17]).…”