Narrow Results

Let $R$ be a ring with identity. The unit (respectively, unitary Cayley) graph of $R$ is a simple graph $G(R)$ (respectively, $\mathrm{\Gamma }(R)$) 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 $\mathrm{\Gamma }(R)$ are isomorphic for a finite ring $R$. Among other results, we prove that $G(R)$ is isomorphic to $\mathrm{\Gamma }(R)$ for a finite ring $R$ if and only if char($R/J(R)$)=2 or $R/J(R)\cong {\mathbb{Z}}_2\times S$, where $J(R)$ is the Jacobson radical of $R$ and $S$ is a finite ring.

The notion of the metric dimension of a graph is well-known and its study is well entrenched in the literature. In this paper, we introduce new classes of graphs that exhibit remarkable characteristic: the metric dimension and the diameter of the unit graph are equal. Additionally, we provide a characterization of finite commutative rings $R$, wherein the metric dimension assumes a value $k$, where $1\le k\le 3$. Further, we also demonstrate an exhaustive examination that ascertains the precise finite commutative rings $R$, in which domination number is equal to metric dimension of unit graph.

Let R be a commutative ring with nonzero identity. For an arbitrary multiplicatively closed subset S of R, we associate a simple graph denoted by Γ_S(R) with all elements of R as vertices, and two distinct vertices x, y ∈ R are adjacent if and only if x+y ∈ S. Two well-known graphs of this type are the total graph and the unit graph. In this paper, we study some basic properties of Γ_S(R). Moreover, we will improve and generalize some results for the total and the unit graphs.

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.

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 S_g. In this paper, we determine all isomorphism classes of finite commutative rings whose unit graphs have genus at most three.

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 $\mathrm{\Gamma }(R)$, the compressed zero-divisor graph ${\mathrm{\Gamma }}_E(R)$ 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 $\mathrm{\Gamma }(R)$ and $\mathrm{\Gamma }(R[x;\alpha ,\delta ])$, as well as the relations between the domination number of $G(R)$ and $G(R[[x;\alpha ]])$, are studied.

Let R be a ring with non-zero identity. The unit graph G(R) of 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 necessarily commutative), then G(R) is a complete r-partite graph if and only if (R, 𝔪) is a local ring and r = |R/𝔪| = 2ⁿ 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.

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.

Let $R$ be a commutative ring and $U(R)$ the multiplicative group of unit elements of $R$. In 2012, Khashyarmanesh et al. defined the generalized unit and unitary Cayley graph, $\Gamma (R,G,S)$, corresponding to a multiplicative subgroup $G$ of $U(R)$ and a nonempty subset $S$ of $G$ with $S^{-1}=\{s^{-1} | s\in S\}\subseteq S$, as the graph with vertex set $R$and two distinct vertices $x$ and $y$ being adjacent if and only if there exists $s\in S$ such that $x+sy\in G$. In this paper, we characterize all Artinian rings $R$ for which $\Gamma (R,U(R),S)$ is projective. This leads us to determine all Artinian rings whose unit graphs, unitary Cayley graphs and co-maximal graphs are projective. In addition, we prove that for an Artinian ring $R$ for which $\Gamma (R,U(R),S)$ has finite nonorientable genus, $R$ must be a finite ring. Finally, it is proved that for a given positive integer $k$, the number of finite rings $R$ for which $\Gamma (R,U(R),S)$ has nonorientable genus $k$ is finite.

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.

In this paper, we consider the unit graph $G({\mathbb{Z}}_n)$, where $n=p_1^{n_1}\text{ or }{p}_1^{n_1}{p}_2^{n_2}\text{ or }{p}_1^{n_1}{p}_2^{n_2}{p}_3^{n_3}$ and $p_1,p_2,p_3$ are distinct primes. For any prime $q$, we construct $q$-ary linear codes from the incidence matrix of the unit graph $G({\mathbb{Z}}_n)$ with their parameters. We also prove that the dual of the constructed codes have minimum distance either three or four. Lastly, we stated two conjectures on diameter of unit graph $G({\mathbb{Z}}_n)$ and linear codes constructed from the incidence matrix of the unit graph $G({\mathbb{Z}}_n)$ for any integer $n$.

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 $G({\mathbb{Z}}_n)$ when $n$ has exactly three prime divisors.