On the Unit Graph of a Non-commutative Ring

Abstract

Let R be a ring (not necessary commutative) with non-zero identity. The unit graph of R, denoted by G(R), is a graph with elements of R as its vertices and two distinct vertices a and b are adjacent if and only if a + b is a unit element of R. It was proved that if R is a commutative ring and m is a maximal ideal of R such that |R/m| = 2, then G(R) is a complete bipartite graph if and only if (R, m) is a local ring. In this paper we generalize this result by showing that if R is a ring (not necessary commutative), then G(R) is a complete r-partite graph if and only if (R, m) is a local ring and r = |R/m| = 2 n , for some n ∈ N or R is a finite field. Among other results we show that if R is a left Artinian ring, 2 ∈ U (R) and the clique number of G(R) is finite, then R is a finite ring.2000 Mathematics Subject Classification. 05C25, 13E10.

Perfect Codes in Induced Subgraph of Unit Graph Associated With Some Commutative Rings

“…In this paper, R indicates the commutative ring with unity, ( ) OR shows the order of , Ashrafi et al [1] defined the unit graph of ring R and proved some results on the properties of the unit graph in terms of connectivity, planarity, girth and diameter. Later, Akbari et al [2] characterized the non-commutative ring having unit graphs as r-partite graph. Furthermore, some other properties of the unit graphs have introduced by many authors which include the Hamiltonian property in [3], dominating number in [4], girth in [5] and the diameter in [6].…”

Graphs Defined on Rings: A Review

“…Additionally, the conditions under which the unit graph of any finite ring was Hamiltonian were obtained in [267] by providing an algorithm that found a spanning cycle of the unit graph, which took the required end points as the inputs and provided the corresponding Hamiltonian cycle. In [268], a short discussion on the unit graphs of non-commutative rings was given, wherein a very few results of the unit graphs of commutative rings were extended by proving them without using the commutative property of the ring. With this study, the challenge to investigate the unit graphs associated with non-commutative rings was clearly visible.…”

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]).…”