Unitary homogeneous bi-Cayley graphs over finite commutative rings

2019 ◽  
Vol 19 (09) ◽  
pp. 2050173
Author(s):  
Xiaogang Liu ◽  
Chengxin Yan
Keyword(s):  
Commutative Ring ◽  
Cayley Graph ◽  
Line Graph ◽  
Cayley Graphs ◽  
Commutative Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Commutative Ring ◽  
Necessary And Sufficient ◽  
Finite Commutative Rings

Let [Formula: see text] denote the unitary homogeneous bi-Cayley graph over a finite commutative ring [Formula: see text]. In this paper, we determine the energy of [Formula: see text] and that of its complement and line graph, and characterize when such graphs are hyperenergetic. We also give a necessary and sufficient condition for [Formula: see text] (respectively, the complement of [Formula: see text], the line graph of [Formula: see text]) to be Ramanujan.

scholarly journals Spectral Properties of Unitary Cayley Graphs of Finite Commutative Rings

10.37236/2390 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Xiaogang Liu ◽  
Sanming Zhou
Keyword(s):  
Spectral Properties ◽  
Line Graph ◽  
Commutative Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Absolute Value ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Unitary Cayley Graph ◽  
Necessary And Sufficient

Let $R$ be a finite commutative ring. The unitary Cayley graph of $R$, denoted $G_R$, is the graph with vertex set $R$ and edge set $\left\{\{a,b\}:a,b\in R, a-b\in R^\times\right\}$, where $R^\times$ is the set of units of $R$. An $r$-regular graph is Ramanujan if the absolute value of every eigenvalue of it other than $\pm r$ is at most $2\sqrt{r-1}$. In this paper we give a necessary and sufficient condition for $G_R$ to be Ramanujan, and a necessary and sufficient condition for the complement of $G_R$ to be Ramanujan. We also determine the energy of the line graph of $G_R$, and compute the spectral moments of $G_R$ and its line graph.

Generalized unit and unitary Cayley graphs of finite rings

2019 ◽  
Vol 18 (01) ◽  
pp. 1950006 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
S. Anukumar Kathirvel
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Unit Element ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Rings ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Necessary And Sufficient ◽  
Nonzero Identity

Let [Formula: see text] be a finite commutative ring with nonzero identity and [Formula: see text] be the set of all units of [Formula: see text] The graph [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] in which two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if there exists a unit element [Formula: see text] in [Formula: see text] such that [Formula: see text] is a unit in [Formula: see text] In this paper, we obtain degree of all vertices in [Formula: see text] and in turn provide a necessary and sufficient condition for [Formula: see text] to be Eulerian. Also, we give a necessary and sufficient condition for the complement [Formula: see text] to be Eulerian, Hamiltonian and planar.

BALANCED UNITARY CAYLEY SIGRAPHS OVER FINITE COMMUTATIVE RINGS

2014 ◽  
Vol 13 (05) ◽  
pp. 1350152 ◽  
Author(s):  
YOTSANAN MEEMARK ◽  
BORWORN SUNTORNPOCH
Keyword(s):  
Cayley Graph ◽  
Line Graph ◽  
Commutative Rings ◽  
Signed Graph ◽  
Group Of Units ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Unitary Cayley Graph ◽  
Finite Commutative Rings

Let R be a finite commutative ring with identity 1. The unitary Cayley graph of R, denoted by GR, is the graph whose vertex set is R and the edge set {{a, b} : a, b ∈ R and a - b ∈ R×}, where R× is the group of units of R. We define the unitary Cayley signed graph (or unitary Cayley sigraph in short) to be an ordered pair 𝒮R = (GR, σ), where GR is the unitary Cayley graph over R with signature σ : E(GR) → {1, -1} given by [Formula: see text] In this paper, we give a criterion on R for SR to be balanced (every cycle in 𝒮R is positive) and a criterion for its line graph L(𝒮R) to be balanced. We characterize all finite commutative rings with the property that the marked sigraph 𝒮R,μ is canonically consistent. Moreover, we give a characterization of all finite commutative rings where 𝒮R, η(𝒮R) and L(𝒮R) are hyperenergetic balanced.

The Square Mapping Graphs of Finite Commutative Rings

2012 ◽  
Vol 19 (03) ◽  
pp. 569-580 ◽  
Author(s):  
Yangjiang Wei ◽  
Gaohua Tang ◽  
Huadong Su
Keyword(s):  
Fixed Points ◽  
Commutative Ring ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Directed Edge ◽  
Induced Subgraph ◽  
Zero Divisors ◽  
Finite Commutative Ring ◽  
Necessary And Sufficient ◽  
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.

scholarly journals Noetherian properties in composite generalized power series rings

2020 ◽  
Vol 18 (1) ◽  
pp. 1540-1551
Author(s):  
Jung Wook Lim ◽  
Dong Yeol Oh
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generalized Power Series ◽  
Ordered Monoid ◽  
Power Series Rings ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

Abstract Let ({\mathrm{\Gamma}},\le ) be a strictly ordered monoid, and let {{\mathrm{\Gamma}}}^{\ast }\left={\mathrm{\Gamma}}\backslash \{0\} . Let D\subseteq E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D. Set \begin{array}{l}D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {E}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(0)\in D\right\}\hspace{.5em}\text{and}\\ \hspace{0.2em}D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {D}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(\alpha )\in I,\hspace{.5em}\text{for}\hspace{.25em}\text{all}\hspace{.5em}\alpha \in {{\mathrm{\Gamma}}}^{\ast }\right\}.\end{array} In this paper, we give necessary conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively ordered, and sufficient conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. Moreover, we give a necessary and sufficient condition for the ring D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. As corollaries, we give equivalent conditions for the rings D+({X}_{1},\ldots ,{X}_{n})E{[}{X}_{1},\ldots ,{X}_{n}] and D+({X}_{1},\ldots ,{X}_{n})I{[}{X}_{1},\ldots ,{X}_{n}] to be Noetherian.

scholarly journals On separable extensions of group rings and quaternion rings

1978 ◽  
Vol 1 (4) ◽  
pp. 433-438
Author(s):  
George Szeto
Keyword(s):  
Commutative Ring ◽  
Full Description ◽  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Ring Extension ◽  
Quaternion Algebras ◽  
Separable Group ◽  
Necessary And Sufficient ◽  
Separable Extensions

The purposes of the present paper are (1) to give a necessary and sufficient condition for the uniqueness of the separable idempotent for a separable group ring extensionRG(Rmay be a non-commutative ring), and (2) to give a full description of the set of separable idempotents for a quaternion ring extensionRQover a ringR, whereQare the usual quaternionsi,j,kand multiplication and addition are defined as quaternion algebras over a field. We shall show thatRGhas a unique separable idempotent if and only ifGis abelian, that there are more than one separable idempotents for a separable quaternion ringRQ, and thatRQis separable if and only if2is invertible inR.

On zero-divisor graphs of commutative rings without identity

2019 ◽  
Vol 19 (12) ◽  
pp. 2050226 ◽  
Author(s):  
G. Kalaimurugan ◽  
P. Vignesh ◽  
T. Tamizh Chelvam
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Zero Divisor Graph ◽  
Finite Commutative Ring ◽  
Free Order ◽  
Finite Commutative Rings

Let [Formula: see text] be a finite commutative ring without identity. In this paper, we characterize all finite commutative rings without identity, whose zero-divisor graphs are unicyclic, claw-free and tree. Also, we obtain all finite commutative rings without identity and of cube-free order for which the corresponding zero-divisor graph is toroidal.

Finite commutative ring with genus two essential graph

2018 ◽  
Vol 17 (07) ◽  
pp. 1850121
Author(s):  
K. Selvakumar ◽  
M. Subajini ◽  
M. J. Nikmehr
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Zero Divisors ◽  
Essential Ideal ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Genus Two ◽  
Finite Commutative Rings

Let [Formula: see text] be a commutative ring with identity and let [Formula: see text] be the set of zero-divisors of [Formula: see text]. The essential graph of [Formula: see text] is defined as the graph [Formula: see text] with the vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is an essential ideal. In this paper, we classify all finite commutative rings with identity for which the genus of [Formula: see text] is two.

PLANAR, OUTERPLANAR, AND RING GRAPH OF THE COZERO-DIVISOR GRAPH OF A FINITE COMMUTATIVE RING

2012 ◽  
Vol 11 (06) ◽  
pp. 1250103 ◽  
Author(s):  
MOJGAN AFKHAMI ◽  
KAZEM KHASHYARMANESH
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Ring Graph ◽  
Nonzero Identity ◽  
Finite Commutative Rings

Let R be a commutative ring with nonzero identity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertex-set W*(R), which is the set of all nonzero and non-unit elements of R, and two distinct vertices a and b in W*(R) are adjacent if and only if a ∉ Rb and b ∉ Ra. In this paper, we characterize all finite commutative rings R such that Γ′(R) is planar, outerplanar or ring graph.

Some results on the complement of the annihilating ideal graph of a commutative ring

2015 ◽  
Vol 14 (07) ◽  
pp. 1550099 ◽  
Author(s):  
S. Visweswaran ◽  
Hiren D. Patel
Keyword(s):  
Commutative Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Rings considered in this article are commutative with identity which admit at least one nonzero annihilating ideal. For such a ring R, we determine necessary and sufficient conditions in order that the complement of its annihilating ideal graph is connected and also find its diameter when it is connected. We discuss the girth of the complement of the annihilating ideal graph of R and prove that it is either equal to 3 or ∞. We also present a necessary and sufficient condition for the complement of the annihilating ideal graph to be complemented.

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