On the Planarity of Graphs Associated with Symmetric and Pseudo Symmetric Numerical Semigroups (original) (raw)

Classification of Planar Graphs Associated to the Ideal of the Numerical Semigroup

2020

Let Λ be a numerical semigroup and I ⊂ Λ be an ideal of Λ. The graph GI(Λ) assigned to an ideal I of Λ is a graph with elements of (Λ \ I) ∗ as vertices and any two vertices x, y are adjacent if and only if x + y ∈ I. In this paper we give a complete characterization (up to isomorphism ) of the graph GI(Λ) to be planar, where I is an irreducible ideal of Λ. This will finally characterize non planar graphs GI (Λ) corresponding to irreducible ideal I.

Characterization of Graphs Associated with Numerical Semigroups

Mathematics

Let Γ be a numerical semigroup. We associate an undirected graph G ( Γ ) with a numerical semigroup Γ with vertex set { v i : i ∈ N \ Γ } and edge set { v i v j ⇔ i + j ∈ Γ } . In this article, we discuss the connectedness, diameter, girth, and some other related properties of the graph G ( Γ ) .

Graphs Associated with the Ideals of a Numerical Semigroup Having Metric Dimension 2

Mathematical Problems in Engineering

Let Λ be a numerical semigroup and I ⊂ Λ be an irreducible ideal of Λ . The graph G I Λ assigned to an ideal I of Λ is a graph with elements of Λ ∖ I ∗ as vertices, and any two vertices x and y are adjacent if and only if x + y ∈ I . In this work, we give a complete characterization (up to isomorphism) of the graph G I Λ having metric dimension 2.

On Delta Sets Of Some Pseudo-Symmetric Numerical Semigroups With Embedding Dimension Three

Bitlis Eren üniversitesi fen bilimleri dergisi, 2022

Let be a numerical semigroup. The catenary degree of an element in is a nonnegative integer used to measure the distance between factorizations of. The catenary degree of the numerical semigroup is obtained at the maximum catenary degree of its elements. The maximum catenary degree of is attained via Betti elements of with complex properties. The Betti elements of can be obtained from all minimal presentations of. A presentation for is a system of generators of the kernel congruence of the special factorization homomorphism. A presentation is minimal if it can not be converted to another presentation, that is, any of its proper subsets is no longer a presentation. The Delta set of is a factorization invariant measuring the complexity of sets of the factorization lengths for the elements in. In this study, we will mainly express the given above invariants of a special pseudosymmetric numerical semigroup family in terms of its generators.

Some properties of Square element graphs over semigroups

AKCE International Journal of Graphs and Combinatorics, 2019

The Square element graph over a semigroup S is a simple undirected graph Sq(S) whose vertex set consists precisely of all the non-zero elements of S, and two vertices a, b are adjacent if and only if either ab or ba belongs to the set {t 2 : t ∈ S} \ {1}, where 1 is the identity of the semigroup (if it exists). In this paper, we study the various properties of Sq(S). In particular, we concentrate on square element graphs over three important classes of semigroups. First, we consider the semigroup Ω n formed by the ideals of Z n. Afterwards, we consider the symmetric groups S n and the dihedral groups D n. For each type of semigroups mentioned, we look into the structural and other graph-theoretic properties of the corresponding square element graphs. c

On a graph of monogenic semigroups

Journal of Inequalities and Applications, 2013

Let us consider the finite monogenic semigroup S M with zero having elements {x, x 2 , x 3 ,. .. , x n }. There exists an undirected graph (S M) associated with S M whose vertices are the non-zero elements x, x 2 , x 3 ,. .. , x n and, f or 1 ≤ i, j ≤ n, any two distinct vertices x i and x j are adjacent if i + j > n. In this paper, the diameter, girth, maximum and minimum degrees, domination number, chromatic number, clique number, degree sequence, irregularity index and also perfectness of (S M) have been established. In fact, some of the results obtained in this section are sharper and stricter than the results presented in DeMeyer et al. (Semigroup Forum 65:206-214, 2002). Moreover, the number of triangles for this special graph has been calculated. In the final part of the paper, by considering two (not necessarily different) graphs (S 1 M) and (S 2 M), we present the spectral properties to the Cartesian product (S 1 M) (S 2 M).

Collected results on semigroups, graphs and codes

2012

In this thesis we present a compendium of _ve works where discrete mathematics play a key role. The _rst three works describe di_erent developments and applications of the semigroup theory while the other two have more independent topics. First we present a result on semigroups and code e_ciency, where we introduce our results on the so-called Geil-Matsumoto bound and Lewittes' bound for algebraic geometry codes. Following that, we work on semigroup ideals and their relation with the Feng-Rao numbers; those numbers, in turn, are used to describe the Hamming weights which are used in a broad spectrum of applications, i.e. the wire-tap channel of type II or in the t-resilient functions used in cryptography. The third work presented describes the non-homogeneous patterns for semigroups, explains three di_erent scenarios where these patterns arise and gives some results on their admissibility. The last two works are not as related as the _rst three but still use discrete mathematics...

Line Graphs of Monogenic Semigroup Graphs

Journal of Mathematics, 2021

e 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)).