On graph associated to co-ideals of commutative semirings

Yahya Talebi; Atefeh Darzi

Displaying similar documents to “On graph associated to co-ideals of commutative semirings”

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.

Some classes of perfect strongly annihilating-ideal graphs associated with commutative rings

Mitra Jalali, Abolfazl Tehranian, Reza Nikandish, Hamid Rasouli (2020)

Commentationes Mathematicae Universitatis Carolinae

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Let $R$ be a commutative ring with identity and $A (R)$ be the set of ideals with nonzero annihilator. The strongly annihilating-ideal graph of $R$ is defined as the graph $SAG (R)$ with the vertex set $A {(R)}^{*} = A (R) ∖ {0}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I \cap Ann (J) \neq (0)$ and $J \cap Ann (I) \neq (0)$ . In this paper, the perfectness of $SAG (R)$ for some classes of rings $R$ is investigated.

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...

The Turán number of the graph $3 P_{4}$

Halina Bielak, Sebastian Kieliszek (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $e x (n, G)$ denote the maximum number of edges in a graph on $n$ vertices which does not contain $G$ as a subgraph. Let $P_{i}$ denote a path consisting of $i$ vertices and let $m P_{i}$ denote $m$ disjoint copies of $P_{i}$ . In this paper we count $e x (n, 3 P_{4})$ .

Some results on the co-intersection graph of submodules of a module

Lotf Ali Mahdavi, Yahya Talebi (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let $R$ be a ring with identity and $M$ be a unitary left $R$ -module. The co-intersection graph of proper submodules of $M$ , denoted by $Ω (M)$ , is an undirected simple graph whose vertex set $V (Ω)$ is a set of all nontrivial submodules of $M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $N + K \neq M$ . We study the connectivity, the core and the clique number of $Ω (M)$ . Also, we provide some conditions on the module $M$ , under which the clique number of $Ω (M)$ is infinite and $Ω (M)$ is a planar graph. Moreover, we give...

Degree sums of adjacent vertices for traceability of claw-free graphs

Tao Tian, Liming Xiong, Zhi-Hong Chen, Shipeng Wang (2022)

Czechoslovak Mathematical Journal

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The line graph of a graph $G$ , denoted by $L (G)$ , has $E (G)$ as its vertex set, where two vertices in $L (G)$ are adjacent if and only if the corresponding edges in $G$ have a vertex in common. For a graph $H$ , define ${\bar{σ}}_{2} (H) = min {d (u) + d (v) : u v \in E (H)}$ . Let $H$ be a 2-connected claw-free simple graph of order $n$ with $δ (H) \geq 3$ . We show that, if ${\bar{σ}}_{2} (H) \geq \frac{1}{7} (2 n - 5)$ and $n$ is sufficiently large, then either $H$ is traceable or the Ryjáček’s closure $cl (H) = L (G)$ , where $G$ is an essentially $2$ -edge-connected triangle-free graph that can be contracted to one of the two graphs of order 10...

The small Ree group $^{2} G_{2} (3^{2 n + 1})$ and related graph

Alireza K. Asboei, Seyed S. S. Amiri (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be a finite group. The main supergraph $𝒮 (G)$ is a graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $o (x) ∣ o (y)$ or $o (y) ∣ o (x)$ . In this paper, we will show that $G ≅^{2} G_{2} (3^{2 n + 1})$ if and only if $𝒮 (G) ≅ 𝒮 (^{2} G_{2} (3^{2 n + 1}))$ . As a main consequence of our result we conclude that Thompson’s problem is true for the small Ree group $^{2} G_{2} (3^{2 n + 1})$ .

Even factor of bridgeless graphs containing two specified edges

Nastaran Haghparast, Dariush Kiani (2018)

Czechoslovak Mathematical Journal

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An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. Let $G$ be a bridgeless simple graph with minimum degree at least $3$ . Jackson and Yoshimoto (2007) showed that $G$ has an even factor containing two arbitrary prescribed edges. They also proved that $G$ has an even factor in which each component has order at least four. Moreover, Xiong, Lu and Han (2009) showed that for each pair of edges $e_{1}$ and $e_{2}$ of $G$ , there is an even factor containing $e_{1}$ and $e_{2}$ ...

Saturation numbers for linear forests $P_{6} + t P_{2}$

Jingru Yan (2023)

Czechoslovak Mathematical Journal

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A graph $G$ is $H$ -saturated if it contains no $H$ as a subgraph, but does contain $H$ after the addition of any edge in the complement of $G$ . The saturation number, $sat (n, H)$ , is the minimum number of edges of a graph in the set of all $H$ -saturated graphs of order $n$ . We determine the saturation number $sat (n, P_{6} + t P_{2})$ for $n \geq \frac{10}{3} t + 10$ and characterize the extremal graphs for $n > \frac{10}{3} t + 20$ .

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...

On sets of discontinuities of functions continuous on all lines

Luděk Zajíček (2022)

Commentationes Mathematicae Universitatis Carolinae

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Answering a question asked by K. C. Ciesielski and T. Glatzer in 2013, we construct a $C^{1}$ -smooth function $f$ on $[0, 1]$ and a closed set $M \subset graph f$ nowhere dense in $graph f$ such that there does not exist any linearly continuous function on $ℝ^{2}$ (i.e., function continuous on all lines) which is discontinuous at each point of $M$ . We substantially use a recent full characterization of sets of discontinuity points of linearly continuous functions on $ℝ^{n}$ proved by T. Banakh and O. Maslyuchenko in 2020. As an easy consequence...

Turán number of two vertex-disjoint copies of cliques

Caiyun Hu (2024)

Czechoslovak Mathematical Journal

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The Turán number of a given graph $H$ , denoted by $ex (n, H)$ , is the maximum number of edges in an $H$ -free graph on $n$ vertices. Applying a well-known result of Hajnal and Szemerédi, we determine the Turán number $ex (n, K_{p} \cup K_{q}$ ) of a vertex-disjoint union of cliques $K_{p}$ and $K_{q}$ for all values of $n$ .

Matchings in complete bipartite graphs and the $r$ -Lah numbers

Gábor Nyul, Gabriella Rácz (2021)

Czechoslovak Mathematical Journal

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We give a graph theoretic interpretation of $r$ -Lah numbers, namely, we show that the $r$ -Lah number ${⌊\binom{n}{k}⌋}_{r}$ counting the number of $r$ -partitions of an $(n + r)$ -element set into $k + r$ ordered blocks is just equal to the number of matchings consisting of $n - k$ edges in the complete bipartite graph with partite sets of cardinality $n$ and $n + 2 r - 1$ ( $0 \leq k \leq n$ , $r \geq 1$ ). We present five independent proofs including a direct, bijective one. Finally, we close our work with a similar result for $r$ -Stirling numbers of the second kind. ...

Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC

Amitayu Banerjee (2023)

Commentationes Mathematicae Universitatis Carolinae

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In set theory without the axiom of choice (AC), we observe new relations of the following statements with weak choice principles. $\circ$ $𝒫_{lf, c}$ (Every locally finite connected graph has a maximal independent set). $\circ$ $𝒫_{lc, c}$ (Every locally countable connected graph has a maximal independent set). $\circ$ CAC $_{1}^{ℵ_{α}}$ (If in a partially ordered set all antichains are finite and all chains have size $ℵ_{α}$ , then the set has size $ℵ_{α}$ ) if $ℵ_{α}$ is regular. $\circ$ CWF (Every partially ordered set has a...

Complete pairs of coanalytic sets

Jean Saint Raymond (2007)

Fundamenta Mathematicae

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Let X be a Polish space, and let C₀ and C₁ be disjoint coanalytic subsets of X. The pair (C₀,C₁) is said to be complete if for every pair (D₀,D₁) of disjoint coanalytic subsets of $ω^{ω}$ there exists a continuous function $f : ω^{ω} \to X$ such that $f^{- 1} (C ₀) = D ₀$ and $f^{- 1} (C ₁) = D ₁$ . We give several explicit examples of complete pairs of coanalytic sets.