scholarly journals Tensor Product Of Zero-divisor Graphs With Finite Free Semilattices

2017 ◽  
Vol 9 (1) ◽  
pp. 13
Author(s):  
Kemal Toker
Keyword(s):  
Tensor Product ◽  
Chromatic Number ◽  
Zero Divisor ◽  
Independence Number ◽  
Domination Number ◽  
Clique Number ◽  
Perfect Graph ◽  
Chromatic Index

$\Gamma (SL_{X})$ is defined and has been investigated in (Toker, 2016). In this paper our main aim is to extend this study over  $\Gamma (SL_{X})$ to the tensor product. The diameter, radius, girth, domination number, independence number, clique number, chromatic number and chromatic index of $\Gamma (SL_{X_{1}})\otimes \Gamma (SL_{X_{2}})$ has been established. Moreover, we have determined when $\Gamma (SL_{X_{1}})\otimes \Gamma (SL_{X_{2}})$ is a perfect graph.

Clean graph of a ring

2020 ◽  
pp. 2150156
Author(s):  
Mohammad HABIBI ◽  
Ece YETKİN ÇELİKEL ◽  
Ci̇hat ABDİOĞLU
Keyword(s):  
Chromatic Number ◽  
Crucial Role ◽  
Independence Number ◽  
Domination Number ◽  
Clique Number ◽  
Idempotent Element ◽  
Nonzero Idempotent

Let [Formula: see text] be a ring (not necessarily commutative) with identity. The clean graph [Formula: see text] of a ring [Formula: see text] is a graph with vertices in form [Formula: see text], where [Formula: see text] is an idempotent and [Formula: see text] is a unit of [Formula: see text]; and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. In this paper, we focus on [Formula: see text], the subgraph of [Formula: see text] induced by the set [Formula: see text] is a nonzero idempotent element of [Formula: see text]. It is observed that [Formula: see text] has a crucial role in [Formula: see text]. The clique number, the chromatic number, the independence number and the domination number of the clean graph for some classes of rings are determined. Moreover, the connectedness and the diameter of [Formula: see text] are studied.

scholarly journals A new graph over semi-direct products of groups

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 611-619
Author(s):  
Sercan Topkaya ◽  
Sinan Cevik
Keyword(s):  
Chromatic Number ◽  
Final Section ◽  
Degree Sequence ◽  
Domination Number ◽  
Clique Number ◽  
Perfect Graph ◽  
Direct Products ◽  
Products Of Groups ◽  
Minimum Degrees ◽  
Product Of Groups

In this paper, by establishing a new graph ?(G) over the semi-direct product of groups, we will first state and prove some graph-theoretical properties, namely, diameter, maximum and minimum degrees, girth, degree sequence, domination number, chromatic number, clique number of ?(G). In the final section we will show that ?(G) is actually a perfect graph.

The Weakly Nilpotent Graph of a Commutative Ring

2017 ◽  
Vol 60 (2) ◽  
pp. 319-328
Author(s):  
Soheila Khojasteh ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Disjoint Union ◽  
Independence Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Unicyclic Graph ◽  
Nilpotent Elements ◽  
Vertex Set

AbstractLet R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

A generalized ideal-based zero-divisor graph

2015 ◽  
Vol 14 (06) ◽  
pp. 1550079 ◽  
Author(s):  
M. J. Nikmehr ◽  
S. Khojasteh
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Chromatic Number ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Clique Number ◽  
Proper Ideal ◽  
Graph Structure ◽  
Zero Divisor Graph ◽  
Vertex Set

Let R be a commutative ring with identity, I its proper ideal and M be a unitary R-module. In this paper, we introduce and study a kind of graph structure of an R-module M with respect to proper ideal I, denoted by ΓI(RM) or simply ΓI(M). It is the (undirected) graph with the vertex set M\{0} and two distinct vertices x and y are adjacent if and only if [x : M][y : M] ⊆ I. Clearly, the zero-divisor graph of R is a subgraph of Γ0(R); this is an important result on the definition. We prove that if ann R(M) ⊆ I and H is the subgraph of ΓI(M) induced by the set of all non-isolated vertices, then diam (H) ≤ 3 and gr (ΓI(M)) ∈ {3, 4, ∞}. Also, we prove that if Spec (R) and ω(Γ Nil (R)(M)) are finite, then χ(Γ Nil (R)(M)) ≤ ∣ Spec (R)∣ + ω(Γ Nil (R)(M)). Moreover, for a secondary R-module M and prime ideal P, we determine the chromatic number and the clique number of ΓP(M), where ann R(M) ⊆ P. Among other results, it is proved that for a semisimple R-module M with ann R(M) ⊆ I, ΓI(M) is a forest if and only if ΓI(M) is a union of isolated vertices or a star.

A ZERO-DIVISOR GRAPH FOR MODULES WITH RESPECT TO THEIR (FIRST) DUAL

2012 ◽  
Vol 12 (02) ◽  
pp. 1250151 ◽  
Author(s):  
M. BAZIAR ◽  
E. MOMTAHAN ◽  
S. SAFAEEYAN
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Complete Graph ◽  
Projective Module ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Clique Number ◽  
Covariant Functor ◽  
Zero Divisor Graph

Let M be an R-module. We associate an undirected graph Γ(M) to M in which nonzero elements x and y of M are adjacent provided that xf(y) = 0 or yg(x) = 0 for some nonzero R-homomorphisms f, g ∈ Hom (M, R). We observe that over a commutative ring R, Γ(M) is connected and diam (Γ(M)) ≤ 3. Moreover, if Γ(M) contains a cycle, then gr (Γ(M)) ≤ 4. Furthermore if ∣Γ(M)∣ ≥ 1, then Γ(M) is finite if and only if M is finite. Also if Γ(M) = ∅, then any nonzero f ∈ Hom (M, R) is monic (the converse is true if R is a domain). For a nonfinitely generated projective module P we observe that Γ(P) is a complete graph. We prove that for a domain R the chromatic number and the clique number of Γ(M) are equal. When R is self-injective, we will also observe that the above adjacency defines a covariant functor between a subcategory of R-MOD and the Category of graphs.

Zero-divisor graphs of matrices over lattices

2016 ◽  
Vol 08 (04) ◽  
pp. 1650060 ◽  
Author(s):  
Anagha Khiste ◽  
Vinayak Joshi
Keyword(s):  
Chromatic Number ◽  
Zero Divisor ◽  
Domination Number ◽  
Basic Properties ◽  
Zero Divisor Graph ◽  
Element Chain ◽  
Text Element ◽  
Triangulated Graph

In this paper, we study basic properties such as connectivity, diameter and girth of the zero-divisor graph [Formula: see text] of [Formula: see text] matrices over a lattice [Formula: see text] with 0. Further, we consider the zero-divisor graph [Formula: see text] of [Formula: see text] matrices over an [Formula: see text]-element chain [Formula: see text]. We determine the number of vertices, degree of each vertex, domination number and edge chromatic number of [Formula: see text]. Also, we show that Beck’s Conjecture is true for [Formula: see text]. Further, we prove that [Formula: see text] is hyper-triangulated graph.

scholarly journals The Non-Crossing Graph

10.37236/1140 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Nathan Linial ◽  
Michael Saks ◽  
David Statter
Keyword(s):  
Chromatic Number ◽  
Independence Number ◽  
Clique Number ◽  
The Other ◽  
Fractional Chromatic Number ◽  
Vertex Set

Two sets are non-crossing if they are disjoint or one contains the other. The non-crossing graph ${\rm NC}_n$ is the graph whose vertex set is the set of nonempty subsets of $[n]=\{1,\ldots,n\}$ with an edge between any two non-crossing sets. Various facts, some new and some already known, concerning the chromatic number, fractional chromatic number, independence number, clique number and clique cover number of this graph are presented. For the chromatic number of this graph we show: $$ n(\log_e n -\Theta(1)) \le \chi({\rm NC}_n) \le n (\lceil\log_2 n\rceil-1). $$

scholarly journals Nonessential sum graph of an Artinian ring

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Bikash Barman ◽  
Kukil Kalpa Rajkhowa
Keyword(s):  
Design Methodology ◽  
Undirected Graph ◽  
Independence Number ◽  
Domination Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Content Type ◽  
Cut Vertex ◽  
Vertex Set ◽  
Regular Character

PurposeThe authors study the interdisciplinary relation between graph and algebraic structure ring defining a new graph, namely “non-essential sum graph”. The nonessential sum graph, denoted by NES(R), of a commutative ring R with unity is an undirected graph whose vertex set is the collection of all nonessential ideals of R and any two vertices are adjacent if and only if their sum is also a nonessential ideal of R.Design/methodology/approachThe method is theoretical.FindingsThe authors obtain some properties of NES(R) related with connectedness, diameter, girth, completeness, cut vertex, r-partition and regular character. The clique number, independence number and domination number of NES(R) are also found.Originality/valueThe paper is original.

scholarly journals Line Graphs of Monogenic Semigroup Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Nihat Akgunes ◽  
Yasar Nacaroglu ◽  
Sedat Pak
Keyword(s):  
Maximum Degree ◽  
Zero Divisor ◽  
Minimum Degree ◽  
Line Graph ◽  
Domination Number ◽  
Clique Number ◽  
Line Graphs ◽  
Monogenic Semigroup ◽  
Graph Properties ◽  
Graph Parameters

The concept of monogenic semigroup graphs Γ S M is firstly introduced by Das et al. (2013) based on zero divisor graphs. In this study, we mainly discuss the some graph properties over the line graph L Γ S M of Γ S M . In detail, we prove the existence of graph parameters, namely, radius, diameter, girth, maximum degree, minimum degree, chromatic number, clique number, and domination number over L Γ S M .

The Strong Perfect Graph Conjecture for Planar Graphs

1973 ◽  
Vol 25 (1) ◽  
pp. 103-114 ◽  
Author(s):  
Alan Tucker
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Independent Set ◽  
Independent Sets ◽  
Clique Number ◽  
Perfect Graph ◽  
Stability Number ◽  
Partition Number ◽  
Minimum Number ◽  
The Stability

A graph G is called γ-perfect if ƛ (H) = γ(H) for every vertex-generated subgraph H of G. Here, ƛ(H) is the clique number of H (the size of the largest clique of H) and γ(H) is the chromatic number of H (the minimum number of independent sets of vertices that cover all vertices of H). A graph G is called α-perfect if α(H) = θ(H) for every vertex-generated subgraph H of G, where α (H) is the stability number of H (the size of the largest independent set of H) and θ(H) is the partition number of H (the minimum number of cliques that cover all vertices of H).

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