A generalization of the unit and unitary Cayley graphs of a commutative ring

Classification of rings with toroidal Jacobson graph

“…In [4], Anderson and Livingston modified and studied the zero-divisor graph Γ(R) as the graph with the nonzero zero-divisors Z(R) * of R as the vertex set. While they focus just on the zero-divisors of the rings (see [1], [2], [3], [4], [10]), there are many other kinds of graphs associated to rings, some of which have been extensively studied, see for example [5], [6], [13], [17], [18].…”

A Refinement of the Unit and Unitary Cayley Graphs of a Finite Ring

“…In [9], Khashayarmanesh and Khorsandi provide a generalization of the unit and unitary Cayley graphs as follows: Let G be a multiplicative subgroup of U (R) and S be a non-empty subset of G such that S −1 = {s −1 | s ∈ S} ⊆ S. Then Γ(R, G, S) is the (simple) graph with vertex set R in which two distinct elements x, y ∈ R are adjacent if and only if there exists s ∈ S such that x+sy ∈ G. The authors in [3] derive several bounds for the genus of Γ(R, U (R), S). In 1198 A. R. NAGHIPOUR AND M. REZAGHOLIBEIGI this paper, we use Γ(R) to denote the graph Γ(R, U (R), U (R)).…”

Representation of finite graphs as difference graphs of S-units, I

“…Independently of these investigations, many people considered the graph G(R, R * ) in the case when R is a finite ring or an Artinian ring. Various properties of these graphs, including connectivity, diameters and chromatic numbers, were studied among others in the papers by Dejter and Giudici [5], Berrizbeitia and Giudici [4], Fuchs [12], Klotz and Sander [31], Lucchini and Maróti [35], Lanski and Maróti [32], Akhtar, Boggess, Jackson-Henderson, Jiménez, Karpman, Kinzel and Pritkin [2] and Khashyarmanesh and Khorsandi [30]. In these works the graph G(R, R * ) is usually called unitary Cayley graph.…”