Displaying similar documents to “Some classes of perfect strongly annihilating-ideal graphs associated with commutative rings”

On graph associated to co-ideals of commutative semirings

Yahya Talebi, Atefeh Darzi (2017)

Commentationes Mathematicae Universitatis Carolinae

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Let R be a commutative semiring with non-zero identity. In this paper, we introduce and study the graph Ω ( R ) whose vertices are all elements of R and two distinct vertices x and y are adjacent if and only if the product of the co-ideals generated by x and y is R . Also, we study the interplay between the graph-theoretic properties of this graph and some algebraic properties of semirings. Finally, we present some relationships between the zero-divisor graph Γ ( R ) and Ω ( R ) .

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 ) ann R ( x ) ann R ( y ) , where for z R , ann R ( z ) = { r 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...

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...

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 10 3 t + 10 and characterize the extremal graphs for n > 10 3 t + 20 .

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 σ ¯ 2 ( H ) = min { d ( u ) + d ( v ) : u v E ( H ) } . Let H be a 2-connected claw-free simple graph of order n with δ ( H ) 3 . We show that, if σ ¯ 2 ( H ) 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 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 ) .

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 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 k n , r 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. ...

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 ...

On g c -colorings of nearly bipartite graphs

Yuzhuo Zhang, Xia Zhang (2018)

Czechoslovak Mathematical Journal

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Let G be a simple graph, let d ( v ) denote the degree of a vertex v and let g be a nonnegative integer function on V ( G ) with 0 g ( v ) d ( v ) for each vertex v V ( G ) . A g c -coloring of G is an edge coloring such that for each vertex v V ( G ) and each color c , there are at least g ( v ) edges colored c incident with v . The g c -chromatic index of G , denoted by χ g c ' ( G ) , is the maximum number of colors such that a g c -coloring of G exists. Any simple graph G has the g c -chromatic index equal to δ g ( G ) or δ g ( G ) - 1 , where δ g ( G ) = min v V ( G ) d ( v ) / g ( v ) . A graph G is nearly bipartite,...

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 𝔭 . We obtain necessary and sufficient conditions on R such that G p ( R ) is disconnected. We find the diameter and...

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 R I : x + y I for some y R I } , where two distinct vertices x and y are adjacent if and only if x + y 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 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 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...

Semi n -ideals of commutative rings

Ece Yetkin Çelikel, Hani A. Khashan (2022)

Czechoslovak Mathematical Journal

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Let R be a commutative ring with identity. A proper ideal I is said to be an n -ideal of R if for a , b R , a b I and a 0 imply b I . We give a new generalization of the concept of n -ideals by defining a proper ideal I of R to be a semi n -ideal if whenever a R is such that a 2 I , then a 0 or a I . We give some examples of semi n -ideal and investigate semi n -ideals under various contexts of constructions such as direct products, homomorphic images and localizations. We present various characterizations of this new...