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.

Normalized Laplacian eigenvalues with chromatic number and independence number of graphs

2018 ◽  
Vol 68 (1) ◽  
pp. 63-80 ◽  
Author(s):  
Shaowei Sun ◽  
Kinkar Ch. Das
Keyword(s):  
Chromatic Number ◽  
Independence Number ◽  
Laplacian Eigenvalues

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.

Spectral gap of a weighted 3-simplicial complex

2021 ◽  
pp. 2150084
Author(s):  
Khalid Hatim ◽  
Azeddine Baalal
Keyword(s):  
Lower Bound ◽  
Simplicial Complex ◽  
Upper Bound ◽  
Spectral Gap ◽  
Simplicial Complexes ◽  
Cheeger Constant ◽  
Nonzero Eigenvalue ◽  
New Framework

In this paper, we construct a new framework that’s we call the weighted [Formula: see text]-simplicial complex and we define its spectral gap. An upper bound for our spectral gap is given by generalizing the Cheeger constant. The lower bound for our spectral gap is obtained from the first nonzero eigenvalue of the Laplacian acting on the functions of certain weighted [Formula: see text]-simplicial complexes.

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.

On distance Laplacian spectrum (energy) of graphs

2020 ◽  
Vol 12 (05) ◽  
pp. 2050061 ◽  
Author(s):  
Hilal A. Ganie
Keyword(s):  
Chromatic Number ◽  
Wiener Index ◽  
Laplacian Spectrum ◽  
Connected Graphs ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
Number Formula ◽  
Energy Formula ◽  
Average Transmission ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] of order [Formula: see text] having distance Laplacian eigenvalues [Formula: see text], the distance Laplacian energy [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the Wiener index of [Formula: see text]. We obtain the distance Laplacian spectrum of the joined union of graphs [Formula: see text] in terms of their distance Laplacian spectrum and the spectrum of an auxiliary matrix. As application, we obtain the distance Laplacian spectrum of the lexicographic product of graphs. We study the distance Laplacian energy of connected graphs with given chromatic number [Formula: see text]. We show that among all connected graphs with chromatic number [Formula: see text] the complete [Formula: see text]-partite graph has the minimum distance Laplacian energy. Further, we discuss the distribution of distance Laplacian eigenvalues around average transmission degree [Formula: see text].

COMBINATORIAL PROPERTIES OF ARNOUX–RAUZY SUBSHIFTS AND APPLICATIONS TO SCHRÖDINGER OPERATORS

2003 ◽  
Vol 15 (07) ◽  
pp. 745-763 ◽  
Author(s):  
D. DAMANIK ◽  
LUCA Q. ZAMBONI
Keyword(s):  
Spectral Theory ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Maximal Power ◽  
One Dimensional ◽  
Combinatorial Properties

We consider Arnoux–Rauzy subshifts X and study various combinatorial questions: When is X linearly recurrent? What is the maximal power occurring in X? What is the number of palindromes of a given length occurring in X? We present applications of our combinatorial results to the spectral theory of discrete one-dimensional Schrödinger operators with potentials given by Arnoux–Rauzy sequences.

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