On the Semigroup Whose Elements Are Subgraphs of a Complete Graph

Journal of Commutative Algebra, 2009

A cut ideal of a graph records the relations among the cuts of the graph. These toric ideals have been introduced by Sturmfels and Sullivant who also posed the problem of relating their properties to the combinatorial structure of the graph. We study the cut ideals of the family of ring graphs, which includes trees and cycles. We show that they have quadratic Gröbner bases and that their coordinate rings are Koszul, Hilbertian, and Cohen-Macaulay, but not Gorenstein in general.

Mathematics and Statistics, 2021

Let S be a commutative ring with unity. The essential ideal graph of S, denoted by E S , is a graph with vertex set consisting of all nonzero proper ideals of A and two vertices P and Q are adjacent whenever P + Q is an essential ideal. An essential ideal P of a ring S is an ideal P of S (P S), having nonzero intersection with every other ideal of S. The set M ax(S) contains all the maximal ideals of S. The Jacobson radical of S, J(S), is the set of intersection of all maximal ideals of S. The comaximal ideal graph of S, denoted by C(S), is a simple graph with vertices as proper ideals of A not contained in J(S) and the vertices P and Q are associated with an edge whenever P + Q = S. In this paper, we study the structural properties of the graph E S by using the ring theoretic concepts. We obtain a characterization for E S to be isomorphic to the comaximal ideal graph C(S). Moreover, we derive the structure theorem of E Zn and determine graph parameters like clique number, chromatic number and independence number. Also, we characterize the perfectness of E Zn and determine the values of n for which E Zn is split and claw-free, Eulerian and Hamiltonian. In addition, we show that the finite essential ideal graph of any non-local ring is isomorphic to E Zn for some n.

Open Mathematics

For a partially ordered set (A,\le ) , let {G}_{A} be the simple, undirected graph with vertex set A such that two vertices a\ne b\in A are adjacent if either a\le b or b\le a . We call {G}_{A} the partial order graph or comparability graph of A. Furthermore, we say that a graph G is a partial order graph if there exists a partially ordered set A such that G={G}_{A} . For a class {\mathcal{C}} of simple, undirected graphs and n, m\ge 1 , we define the Ramsey number { {\mathcal R} }_{{\mathcal{C}}}(n,m) with respect to {\mathcal{C}} to be the minimal number of vertices r such that every induced subgraph of an arbitrary graph in {\mathcal{C}} consisting of r vertices contains either a complete n-clique {K}_{n} or an independent set consisting of m vertices. In this paper, we determine the Ramsey number with respect to some classes of partial order graphs. Furthermore, some implications of Ramsey numbers in ring theory are discussed.

2013

Let R be a commutative ring with nonzero identity and H be a nonempty proper subset of R such that R\H is a saturated multiplicatively closed subset of R. The generalized total graph of R is the (simple) graph GT H (R) with all elements of R as the vertices, and two distinct vertices x and y are adjacent if and only if x + y ∈ H. In this paper, we investigate the structure of GT H (R).

2016

Let R be a commutative ring with identity and A(R) be the set of ideals of R with non-zero annihilators. In this paper, we first introduce and investigate the principal ideal subgraph of the annihilating-ideal graph of R, denoted by AGP (R). It is a (undirected) graph with vertices AP (R) = A(R) ∩ P(R) \ {(0)}, where P(R) is the set of proper principal ideals of R and two distinct vertices I and J are adjacent if and only if IJ = (0). Then, we study some basic properties of AGP (R). For instance, we characterize rings for which AGP (R) is finite graph, complete graph, bipartite graph or star graph. Also, we study diameter and girth of AGP (R). Finally, we compare the principal ideal subgraph AGP (R) and spectrum subgraph AGs(R).

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.

2017

Let S be a non trivial additive cancellation commutative semi-group having a zero 0 with the quotient group G = {s 1 − s 2 | s 1 , s 2 ∈ S}. A prime ideal P of S is called strongly prime if for a, b ∈ G, whenever a + b ∈ P , we get a ∈ P or b ∈ P. We show that a prime ideal of S is strongly prime ⇔ it is powerful. Finally, we prove that if O is an oversemigroup of S and S and O have the nonzero ideal I where I is powerful in O, then 3I is a powerful ideal of S.

The rings considered in this article are commutative with identity which admit at least two maximal ideals. Let R be a ring such that R admits at least two maximal ideals. Recall from Ye and Wu (J. Algebra Appl. 11(6): 1250114, 2012) that the comaximal ideal graph of R, denoted by C (R) is an undirected simple graph whose vertex set is the set of all proper ideals I of R such that I ̸⊆ J(R), where J(R) is the Jacobson radical of R and distinct vertices I1, I2 are joined by an edge in C(R) if and only if I1 + I2 = R. In Section 2 of this article, we classify rings R such that C (R) is planar. In Section 3 of this article, we classify rings R such that C (R) is a split graph. In Section 4 of this article, we classify rings R such that C(R) is complemented and moreover, we determine the S-vertices of C (R).

2008

In a paper in 2008, Herzog, Hibi and Ohsugi introduced and studied the semigroup ring associated to the set of minimal vertex covers of an unmixed bipartite graph. In this paper we relate the dimension of this semigroup ring to the rank of the Boolean lattice associated to the graph. * email cristina.bertone@unito.it † email vmicale@dmi.unict.it 0

Journal of Combinatorial Theory, 1968

If M is a maximal (proper) subsemigroup of a finite semigroup S, then M contains all but one J-class J(M) of S. When J(M)is non-regular J(M)n M = ~ so M = S --J(M). When J(M) is regular either J(M) c3 M = ~ or M n J(M) has a natural form with respect to the Green-Rees coordinates in J(M). Specifically, there exist an isomorphism j : J(M) ~ ~ dl~ B, G, C) of J(M) ~ with a Rees regular matrix semigroup so that j(M c~ J(M)) = G" x A x B, where G' is a maximal subgroup of G or j(M n J(M)) is the complement of a "rectangle" of ,,~-classes of.Z/~ B, G, C). In the first case, (M c3 J(M)) o is a maximal subsemigroup of J(M) ~ In the second, (M n J(M)) o is maximal in J(M) ~ when j(M n J(M)) = r162176 B, G, C) --(G • A' x B ~) for proper subsets A' and B' of A and B, respectively, but need not be when j(Mn J(M)) = G X A X B' or j(M n J(M)) = G X A' X B. The notation of this paper, with slight variations, follows [1]. ~'~ B, G, C) denotes a Rees matrix semigroup with finite index sets A, B, finite group G and C : B • A ~ G o the structure matrix. If J is a J-class of a semigroup S, we denote by j0 the semigroup J w {0}, 0 r J, with multiplication tjlj~ ,