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Let R be a ring with identity. The unit (respectively, unitary Cayley) graph of R is a simple graph G(R) (respectively, Γ(R)) with vertex set R, where two distinct vertices x and y are adjacent if and only if x + y (respectively, x − y) is a unit of R. In this paper, we explore when G(R) and Γ(R) are isomorphic for a finite ring R. Among other results, we prove that G(R) is isomorphic to Γ(R) for a finite ring R if and only if char(R/J(R))=2 or R/J(R)≅ℤ2 × S, where J(R) is the Jacobson radical of R and S is a finite ring. [ABSTRACT FROM AUTHOR]

Let R be a ring with non-zero identity. The unit graph of R , denoted by G (R) , is an undirected graph with all the elements of R as vertices and where distinct vertices x , y are adjacent if and only if x + y is a unit of R. In this paper, we investigate the Wiener index and hyper-Wiener index of G (R) and explicitly determine their values. [ABSTRACT FROM AUTHOR]

The unit graph of a ring R is the simple graph, denoted by G (R) , whose vertex set is R , and in which two distinct vertices x and y are adjacent if and only if x + y is a unit of R. In this paper, we completely determine the domination number of G (ℤ n) when n has exactly three prime divisors. [ABSTRACT FROM AUTHOR]

The unit graph of a ring is the simple graph whose vertices are the elements of the ring and where two distinct vertices are adjacent if and only if their sum is a unit of the ring. A simple graph is said to be planar if it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In this note, we completely characterize the semipotent rings whose unit graphs are planar. As a consequence, we list all semilocal rings with planar unit graphs. [ABSTRACT FROM AUTHOR]

The present work aims to exploit the interplay between the algebraic properties of rings and the graph-theoretic structures of their associated graphs. Let R be an associative (not necessarily commutative) ring. We focus on the domination number of the zero-divisor graph Γ (R) , the compressed zero-divisor graph Γ E (R) and the unit graph G (R). We find some relations between the domination number of the zero-divisor graph and that of the compressed zero-divisor graph. Moreover, some relations between the domination number of Γ (R) and Γ (R [ x ; α , δ ]) , as well as the relations between the domination number of G (R) and G (R [ [ x ; α ] ]) , are studied. [ABSTRACT FROM AUTHOR]

Let 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. [ABSTRACT FROM AUTHOR]

The unit graph of a ring R, denoted G(R), is the simple graph defined on the elements of R with an edge between vertices x and y iff x + y is a unit of R. In this paper, we prove that the girth (G(R)) of the unit graph of an arbitrary ring R is 3, 4, 6 or ∞. We determine the rings R with R/J(R) semipotent and with (G(R)) = 6 or ∞, and classify the rings R with R/J(R) right self-injective and with (G(R)) = 3 or 4. [ABSTRACT FROM AUTHOR]