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.

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.

scholarly journals Bipartite graphs with close domination and k-domination numbers

2020 ◽  
Vol 18 (1) ◽  
pp. 873-885
Author(s):  
Gülnaz Boruzanlı Ekinci ◽  
Csilla Bujtás
Keyword(s):  
Positive Integer ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{Δ}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

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.

When is the Jacobson graph projective?

2018 ◽  
Vol 17 (09) ◽  
pp. 1850168
Author(s):  
Atossa Parsapour ◽  
Khadijeh Ahmadjavaheri
Keyword(s):  
Commutative Ring ◽  
Jacobson Radical ◽  
Jacobson Graph ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Nonzero Identity

Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] be the Jacobson radical of [Formula: see text]. The Jacobson graph of [Formula: see text], denoted by [Formula: see text], is a graph with vertex-set [Formula: see text], such that two distinct vertices [Formula: see text] and [Formula: see text] in [Formula: see text] are adjacent if and only if [Formula: see text] is not a unit of [Formula: see text]. The goal in this paper is to list every finite commutative ring [Formula: see text] with nonzero identity (up to isomorphism) such that the graph [Formula: see text] is projective.

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.

scholarly journals Degree Constrained Orientations in Countable Graphs

10.37236/846 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Attila Bernáth ◽  
Henning Bruhn
Keyword(s):  
Undirected Graph ◽  
Upper Bounds ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Lower And Upper Bounds ◽  
Finite Graphs ◽  
Degree Function ◽  
Necessary And Sufficient

Degree constrained orientations are orientations of an (undirected) graph where the in-degree function satisfies given lower and upper bounds. For finite graphs Frank and Gyárfás (1976) gave a necessary and sufficient condition for the existence of such an orientation. We extend their result to countable graphs.

scholarly journals Commutative feebly nil-clean group rings

2019 ◽  
Vol 11 (2) ◽  
pp. 264-270
Author(s):  
Peter V. Danchev
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Clean Rings ◽  
Unital Ring ◽  
Necessary And Sufficient

Abstract An arbitrary unital ring R is called feebly nil-clean if any its element is of the form q + e − f, where q is a nilpotent and e, f are idempotents with ef = fe. For any commutative ring R and any abelian group G, we find a necessary and sufficient condition when the group ring R(G) is feebly nil-clean only in terms of R, G and their sections. Our result refines establishments due to McGovern et al. in J. Algebra Appl. (2015) on nil-clean rings and Danchev-McGovern in J. Algebra (2015) on weakly nil-clean rings, respectively.

Comaximal factorization of lifting ideals

2020 ◽  
pp. 2250044
Author(s):  
Esmaeil Rostami ◽  
Sina Hedayat ◽  
Reza Nekooei ◽  
Somayeh Karimzadeh
Keyword(s):  
Commutative Ring ◽  
Proper Ideal ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
New Approach ◽  
Ideal Formula ◽  
Basic Properties ◽  
Necessary And Sufficient

A proper ideal [Formula: see text] of a commutative ring [Formula: see text] is called lifting whenever idempotents of [Formula: see text] lift to idempotents of [Formula: see text]. In this paper, many of the basic properties of lifting ideals are studied and we prove and extend some well-known results concerning lifting ideals and lifting idempotents by a new approach. Furthermore, we give a necessary and sufficient condition for every proper ideal of a commutative ring to be a product of pairwise comaximal lifting ideals.

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