A graph associated to proper non-small ideals of a commutative ring

S. Ebrahimi Atani; S. Dolati Pish Hesari; M. Khoramdel

Displaying similar documents to “A graph associated to proper non-small ideals of a commutative ring”

The prime ideals intersection graph of a ring

M. J. Nikmehr, B. Soleymanzadeh (2017)

Commentationes Mathematicae Universitatis Carolinae

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Let $R$ be a commutative ring with unity and $U (R)$ be the set of unit elements of $R$ . In this paper, we introduce and investigate some properties of a new kind of graph on the ring $R$ , namely, the prime ideals intersection graph of $R$ , denoted by $G_{p} (R)$ . The $G_{p} (R)$ is a graph with vertex set $R^{*} - U (R)$ and two distinct vertices $a$ and $b$ are adjacent if and only if there exists a prime ideal $𝔭$ of $R$ such that $a, b \in 𝔭$ . We obtain necessary and sufficient conditions on $R$ such that $G_{p} (R)$ is disconnected. We find the diameter and...

Some results on the annihilator graph of a commutative ring

Mojgan Afkhami, Kazem Khashyarmanesh, Zohreh Rajabi (2017)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring. The annihilator graph of $R$ , denoted by $AG (R)$ , is the undirected graph with all nonzero zero-divisors of $R$ as vertex set, and two distinct vertices $x$ and $y$ are adjacent if and only if ${ann}_{R} (x y) \neq {ann}_{R} (x) \cup {ann}_{R} (y)$ , where for $z \in R$ , ${ann}_{R} (z) = {r \in R : r z = 0}$ . In this paper, we characterize all finite commutative rings $R$ with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings $R$ whose annihilator graphs have clique number $1$ , $2$ or $3$ . Also, we investigate some properties...

A co-ideal based identity-summand graph of a commutative semiring

S. Ebrahimi Atani, S. Dolati Pish Hesari, M. Khoramdel (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $I$ be a strong co-ideal of a commutative semiring $R$ with identity. Let $Γ_{I} (R)$ be a graph with the set of vertices $S_{I} (R) = {x \in R ∖ I : x + y \in I$ for some $y \in R ∖ I}$ , where two distinct vertices $x$ and $y$ are adjacent if and only if $x + y \in I$ . We look at the diameter and girth of this graph. Also we discuss when $Γ_{I} (R)$ is bipartite. Moreover, studies are done on the planarity, clique, and chromatic number of this graph. Examples illustrating the results are presented.

Domination and independence subdivision numbers of graphs

Teresa W. Haynes, Sandra M. Hedetniemi, Stephen T. Hedetniemi (2000)

Discussiones Mathematicae Graph Theory

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The domination subdivision number $s d_{γ} (G)$ of a graph is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number. Arumugam showed that this number is at most three for any tree, and conjectured that the upper bound of three holds for any graph. Although we do not prove this interesting conjecture, we give an upper bound for the domination subdivision number for any graph G in terms of the minimum degrees of...

On 𝓕-independence in graphs

Frank Göring, Jochen Harant, Dieter Rautenbach, Ingo Schiermeyer (2009)

Discussiones Mathematicae Graph Theory

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Let be a set of graphs and for a graph G let $α_{} (G)$ and $α *_{} (G)$ denote the maximum order of an induced subgraph of G which does not contain a graph in as a subgraph and which does not contain a graph in as an induced subgraph, respectively. Lower bounds on $α_{} (G)$ and $α *_{} (G)$ are presented.

Classification of rings with toroidal Jacobson graph

Krishnan Selvakumar, Manoharan Subajini (2016)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with nonzero identity and $J (R)$ the Jacobson radical of $R$ . The Jacobson graph of $R$ , denoted by $𝔍_{R}$ , is defined as the graph with vertex set $R ∖ J (R)$ such that two distinct vertices $x$ and $y$ are adjacent if and only if $1 - x y$ is not a unit of $R$ . The genus of a simple graph $G$ is the smallest nonnegative integer $n$ such that $G$ can be embedded into an orientable surface $S_{n}$ . In this paper, we investigate the genus number of the compact Riemann surface in which $𝔍_{R}$ can be embedded and...

Comaximal graph of $C (X)$

Mehdi Badie (2016)

Commentationes Mathematicae Universitatis Carolinae

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In this article we study the comaximal graph $Γ_{_{2}}^{'} C (X)$ of the ring $C (X)$ . We have tried to associate the graph properties of $Γ_{_{2}}^{'} C (X)$ , the ring properties of $C (X)$ and the topological properties of $X$ . Radius, girth, dominating number and clique number of the $Γ_{_{2}}^{'} C (X)$ are investigated. We have shown that $2 \leq Rad Γ_{_{2}}^{'} C (X) \leq 3$ and if $| X | > 2$ then $girth Γ_{_{2}}^{'} C (X) = 3$ . We give some topological properties of $X$ equivalent to graph properties of $Γ_{_{2}}^{'} C (X)$ . Finally we have proved that $X$ is an almost $P$ -space which does not have isolated points if and only if $C (X)$ is an almost...

Nonempty intersection of longest paths in a graph with a small matching number

Fuyuan Chen (2015)

Czechoslovak Mathematical Journal

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A maximum matching of a graph $G$ is a matching of $G$ with the largest number of edges. The matching number of a graph $G$ , denoted by $α^{'} (G)$ , is the number of edges in a maximum matching of $G$ . In 1966, Gallai conjectured that all the longest paths of a connected graph have a common vertex. Although this conjecture has been disproved, finding some nice classes of graphs that support this conjecture is still very meaningful and interesting. In this short note, we prove that Gallai’s conjecture...

Some remarks on α-domination

Franz Dahme, Dieter Rautenbach, Lutz Volkmann (2004)

Discussiones Mathematicae Graph Theory

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Let α ∈ (0,1) and let $G = (V_{G}, E_{G}$ ) be a graph. According to Dunbar, Hoffman, Laskar and Markus [3] a set $D \subseteq V_{G}$ is called an α-dominating set of G, if $| N_{G} (u) \cap D | \geq α d_{G} (u)$ for all $u \in V_{G} ∖ D$ . We prove a series of upper bounds on the α-domination number of a graph G defined as the minimum cardinality of an α-dominating set of G.

On the diameter of the intersection graph of a finite simple group

Xuanlong Ma (2016)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group. The intersection graph $Δ_{G}$ of $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of $G$ , and two distinct vertices $X$ and $Y$ are adjacent if $X \cap Y \neq 1$ , where $1$ denotes the trivial subgroup of order $1$ . A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters...

Semiproper ideals

Hiroshi Sakai (2005)

Fundamenta Mathematicae

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We say that an ideal I on $_{κ} λ$ is semiproper if the corresponding poset $ℙ_{I}$ is semiproper. In this paper we investigate properties of semiproper ideals on $_{κ} λ$ .

On the order of certain close to regular graphs without a matching of given size

Sabine Klinkenberg, Lutz Volkmann (2007)

Czechoslovak Mathematical Journal

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A graph $G$ is a ${d, d + k}$ -graph, if one vertex has degree $d + k$ and the remaining vertices of $G$ have degree $d$ . In the special case of $k = 0$ , the graph $G$ is $d$ -regular. Let $k, p \geq 0$ and $d, n \geq 1$ be integers such that $n$ and $p$ are of the same parity. If $G$ is a connected ${d, d + k}$ -graph of order $n$ without a matching $M$ of size $2 | M | = n - p$ , then we show in this paper the following: If $d = 2$ , then $k \geq 2 (p + 2)$ and (i) $n \geq k + p + 6$ . If $d \geq 3$ is odd and $t$ an integer with $1 \leq t \leq p + 2$ , then (ii) $n \geq d + k + 1$ for $k \geq d (p + 2)$ , (iii) $n \geq d (p + 3) + 2 t + 1$ for $d (p + 2 - t) + t \leq k \leq d (p + 3 - t) + t - 3$ , (iv) $n \geq d (p + 3) + 2 p + 7$ for $k \leq p$ . If $d \geq 4$ is even, then (v) $n \geq d + k + 2 - η$ for $k \geq d (p + 3) + p + 4 + η$ , (vi) $n \geq d + k + p + 2 - 2 t = d (p + 4) + p + 6$ for $k = d (p + 3) + 4 + 2 t$ and $p \geq 1$ ,...

On the total k-domination number of graphs

Adel P. Kazemi (2012)

Discussiones Mathematicae Graph Theory

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Let k be a positive integer and let G = (V,E) be a simple graph. The k-tuple domination number $γ_{\times k} (G)$ of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V, $| N_{G} [v] \cap S | \geq k$ . Also the total k-domination number $γ_{\times k, t} (G)$ of G is the minimum cardinality of a total k -dominating set S, a set that for every vertex v ∈ V, $| N_{G} (v) \cap S | \geq k$ . The k-transversal number τₖ(H) of a hypergraph H is the minimum size of a subset S ⊆ V(H) such that |S ∩e | ≥ k for every edge e ∈ E(H). We know that for...

Intersection graph of gamma sets in the total graph

T. Tamizh Chelvam, T. Asir (2012)

Discussiones Mathematicae Graph Theory

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In this paper, we consider the intersection graph $I_{Γ} (ℤ ₙ)$ of gamma sets in the total graph on ℤₙ. We characterize the values of n for which $I_{Γ} (ℤ ₙ)$ is complete, bipartite, cycle, chordal and planar. Further, we prove that $I_{Γ} (ℤ ₙ)$ is an Eulerian, Hamiltonian and as well as a pancyclic graph. Also we obtain the value of the independent number, the clique number, the chromatic number, the connectivity and some domination parameters of $I_{Γ} (ℤ ₙ)$ .

On domains with ACC on invertible ideals

Stefania Gabelli (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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If $A$ is a domain with the ascending chain condition on (integral) invertible ideals, then the group $I(A)$ of its invertible ideals is generated by the set $I_{m}(A)$ of maximal invertible ideals. In this note we study some properties of $I_{m}(A)$ and we prove that, if $I(A)$ is a free group on $I_{m}(A)$ , then $A$ is a locally factorial Krull domain.

On norm closed ideals in $L (ℓ_{p}, ℓ_{q})$

B. Sari, Th. Schlumprecht, N. Tomczak-Jaegermann, V. G. Troitsky (2007)

Studia Mathematica

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It is well known that the only proper non-trivial norm closed ideal in the algebra L(X) for $X = ℓ_{p}$ (1 ≤ p < ∞) or X = c₀ is the ideal of compact operators. The next natural question is to describe all closed ideals of $L (ℓ_{p} \oplus ℓ_{q})$ for 1 ≤ p,q < ∞, p ≠ q, or equivalently, the closed ideals in $L (ℓ_{p}, ℓ_{q})$ for p < q. This paper shows that for 1 < p < 2 < q < ∞ there are at least four distinct proper closed ideals in $L (ℓ_{p}, ℓ_{q})$ , including one that has not been studied before. The proofs use various methods...