Huadong Su

Commutative Zero-divisor Semigroups of Graphs with at Most Four Vertices

The zero-divisor graph of a commutative semigroup with zero is a graph whose vertices are the nonzero zero-divisors of the semigroup, with two distinct vertices joined by an edge in case their product in the semigroup is zero. In this paper, we study commutative zero-divisor semigroups determined by graphs. We determine all corresponding zero-divisor semigroups of all simple graphs with at most four vertices.

The Square Mapping Graphs of Finite Commutative Rings

For a finite commutative ring R, the square mapping graph of R is a directed graph Γ(R) whose set of vertices is all the elements of R and for which there is a directed edge from a to b if and only if a2=b. We establish necessary and sufficient conditions for the existence of isolated fixed points, and the cycles with length greater than 1 in Γ(R). We also examine when the induced subgraph on the set of zero-divisors of a local ring with odd characteristic is semiregular. Moreover, we completely determine the finite commutative rings whose square mapping graphs have exactly two, three or four components.

Matrices over a commutative ring as sums of three idempotents or three involutions

Abstract Motivated by Hirano-Tominaga’s work on rings for which every element is a sum of two idempotents and by de Seguins Pazzis’s results on decomposing every matrix over a field of positive characteristic as a sum of idempotent matrices, we address decomposing every matrix over a commutative ring as a sum of three idempotent matrices and, respectively, as a sum of three involutive matrices.

ON THE GIRTH OF THE UNIT GRAPH OF A RING

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 gr(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 gr(G(R)) = 6 or ∞, and classify the rings R with R/J(R) right self-injective and with gr(G(R)) = 3 or 4.

A characterization of rings whose unitary Cayley graphs are planar

Let R be a ring with identity. The unitary Cayley graph of R is the simple graph with vertex set R, where two distinct vertices x and y are linked by an edge if and only if x − y is a unit of R. A 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 paper, we completely characterize the rings whose unitary Cayley graphs are planar.

Units on the Gauss extension of a Galois ring

Ring extensions are a well-studied topic in ring theory. In this paper, we study the structure of the Gauss extension of a Galois ring. We determine the structures of the extension ring and its unit group.

On the Diameter of Unitary Cayley Graphs of Rings

_e unitary Cayley graph of a ring R, denoted Γ(R), is the simple graph deûned on all elements of R, and where two vertices x and y are adjacent if and only if x − y is a unit in R. _e largest distance between all pairs of vertices of a graph G is called the diameter of G, and is denoted by diam(G). It is proved that for each integer n ≥ 1, there exists a ring R such that diam(Γ(R)) = n. We also show that diam(Γ(R)) ∈ {1, 2, 3,∞} for a ring Rwith R/J(R) self-injective and classify all those rings with diam(Γ(R)) = 1, 2, 3 and∞, respectively. _is extends [?, _eorem 3.1].

Finite commutative rings with higher genus unit graphs

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.

On the Genus of the Zero-Divisor Graph of Zn

Let be a commutative ring with identity. The zero-divisor graph of , denoted , is the simple graph whose vertices are the nonzero zero-divisors of , and two distinct vertices and are linked by an edge if and only if . The genus of a simple graph is the smallest integer such that can be embedded into an orientable surface . In this paper, we determine that the genus of the zero-divisor graph of , the ring of integers modulo , is two or three.