Approximating the Independence Number and the Chromatic Number in Expected Polynomial Time

2000 ◽  
pp. 13-24 ◽  
Author(s):  
Michael Krivelevich ◽  
H. Van Vu
Keyword(s):  
Chromatic Number ◽  
Polynomial Time ◽  
Independence Number

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.

THE JACOBSON GRAPH OF COMMUTATIVE RINGS

2012 ◽  
Vol 12 (03) ◽  
pp. 1250179 ◽  
Author(s):  
A. AZIMI ◽  
A. ERFANIAN ◽  
M. FARROKHI D. G.
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Independence Number ◽  
Commutative Rings ◽  
Dominating Number ◽  
Jacobson Graph ◽  
Vertex Set ◽  
Numerical Invariants ◽  
Nonzero Identity

Let R be a commutative ring with nonzero identity. Then the Jacobson graph of R, denoted by 𝔍R, is defined as a graph with vertex set R\J(R) such that two distinct vertices x and y are adjacent if and only if 1 - xy is not a unit of R. We obtain some graph theoretical properties of 𝔍R including its connectivity, planarity and perfectness and we compute some of its numerical invariants, namely diameter, girth, dominating number, independence number and vertex chromatic number and give an estimate for its edge chromatic number.

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 Determining the chromatic number of triangle-free 2P3-free graphs in polynomial time

2012 ◽  
Vol 423 ◽  
pp. 1-10 ◽  
Author(s):  
Hajo Broersma ◽  
Petr A. Golovach ◽  
Daniël Paulusma ◽  
Jian Song
Keyword(s):  
Chromatic Number ◽  
Polynomial Time

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 Enumeration and Asymptotic Properties of Unlabeled Outerplanar Graphs

10.37236/984 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Manuel Bodirsky ◽  
Éric Fusy ◽  
Mihyun Kang ◽  
Stefan Vigerske
Keyword(s):  
Chromatic Number ◽  
Polynomial Time ◽  
Asymptotic Properties ◽  
Statistical Properties ◽  
Cycle Index ◽  
Outerplanar Graphs ◽  
Analytic Combinatorics ◽  
Exact Number ◽  
Asymptotic Number

We determine the exact and asymptotic number of unlabeled outerplanar graphs. The exact number $g_{n}$ of unlabeled outerplanar graphs on $n$ vertices can be computed in polynomial time, and $g_{n}$ is asymptotically $g\, n^{-5/2}\rho^{-n}$, where $g\approx0.00909941$ and $\rho^{-1}\approx7.50360$ can be approximated. Using our enumerative results we investigate several statistical properties of random unlabeled outerplanar graphs on $n$ vertices, for instance concerning connectedness, the chromatic number, and the number of edges. To obtain the results we combine classical cycle index enumeration with recent results from analytic combinatorics.

scholarly journals Spectrum of Signless 1-Laplacian on Simplicial Complexes

10.37236/8951 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Xin Luo ◽  
Dong Zhang
Keyword(s):  
Chromatic Number ◽  
Spectral Theory ◽  
Independence Number ◽  
Simplicial Complexes ◽  
Nodal Domain ◽  
Largest Eigenvalue ◽  
Cheeger Constant ◽  
Laplacian Eigenvalues ◽  
Combinatorial Properties ◽  
Sandwich Theorem

We introduce the signless 1-Laplacian and the dual Cheeger constant on simplicial complexes.  The connection of its spectrum to the combinatorial properties like independence number,  chromatic number and dual Cheeger constant is investigated. Our estimates  can be comparable to Hoffman's bounds on Laplacian eigenvalues of simplicial complexes. An interesting inequality involving multiplicity of the largest eigenvalue, independence number and chromatic number is provided, which could be regarded as a variant version of Lovász sandwich theorem. Also, the behavior of 1-Laplacian under the topological operations of wedge and duplication of motifs is studied. The Courant nodal domain theorem in spectral theory is extended to the setting of signless 1-Laplacian on complexes.

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.

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