Characterization of Graphs Associated with Numerical Semigroups (original) (raw)

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 the Planarity of Graphs Associated with Symmetric and Pseudo Symmetric Numerical Semigroups

Mathematics

Let S(m,e) be a class of numerical semigroups with multiplicity m and embedding dimension e. We call a graph GS an S(m,e)-graph if there exists a numerical semigroup S∈S(m,e) with V(GS)={x:x∈g(S)} and E(GS)={xy⇔x+y∈S}, where g(S) denotes the gap set of S. The aim of this article is to discuss the planarity of S(m,e)-graphs for some cases where S is an irreducible numerical semigroup.

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.

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).

IJERT-Some Properties of Semigraph and its Associated Graphs

International Journal of Engineering Research and Technology (IJERT), 2014

https://www.ijert.org/some-properties-of-semigraph-and-its-associated-graphs https://www.ijert.org/research/some-properties-of-semigraph-and-its-associated-graphs-IJERTV3IS051169.pdf In this paper, some properties of degrees of vertices of a semigraph have been discussed. Four types of graphs associated to a given semigraph have been defined, and size of each graph has also been discussed.

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

Gapsets and numerical semigroups

Journal of Combinatorial Theory, Series A

For g ≥ 0, let n g denote the number of numerical semigroups of genus g. A conjecture by Maria Bras-Amorós in 2008 states that the inequality n g ≥ n g−1 + n g−2 should hold for all g ≥ 2. Here we show that such an inequality holds for the very large subtree of numerical semigroups satisfying c ≤ 3m, where c and m are the conductor and multiplicity, respectively. Our proof is given in the more flexible setting of gapsets, i.e. complements in N of numerical semigroups.

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)).