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![Figure 4. The unit graph G ( ޚ 3 [ x ] / x 2 ) and its 4-colouring.](profile/M-R-Pournaki/publication/228839295/figure/fig4/AS:300828255637505@1448734453048/The-unit-graph-G-kh-3-x-x-2-and-its-4-colouring_Q320.jpg)
A graph is called weakly perfect if its chromatic number equals its clique number. In this paper a new class of weakly perfect graphs arising from rings are presented and an explicit formula for the chromatic number of such graphs is given.
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Context 1
... us note that the proof of Theorem 2.2 gives us an explicit formula for the chromatic number of G ( R ). For a given ring R , we may write R ∼ R × . . . × R , where Note that every this R i formula is a local reduced ring with to the maximal simpler ideal form m if i . all If of 2 ∈ / the U ( R R i ), ’s are by reordering, fields. we may assume that R 1 , . . . , R all ⎧ have characteristic equal to 2 with | R 1 | / | m 1 | ≤ . . . ≤ | R | / | m | and R + 1 , . . . , R n all have characteristic n not equal to 2. Then χ ( G ( R )) = ⎧ ⎨ ⎪ 2 1 n ( | R i | − 1) + n , if 2 ∈ U ( R ) , ⎩ ⎪ n i = 1 χ ( G ( R )) = ⎪ ⎨ 2 1 | n R 1 | , ( | R i | − | m i | ) + n , if if 2 2 ∈ / ∈ U U ( R ( R ) . ) , We conclude this paper with ⎪ ⎩ two i = 1 examples which illustrate the main result and the above formulas. | R | / | m | , if 2 ∈ / U ( R ) . E XAMPLE 3.9. Let k > 1 be an integer and write k = p r 1 1 . . . p r n n , where the p i ’s are distinct prime numbers and the r i ’s are positive integers. Therefore, we obtain ޚ k ∼ = ޚ p r 1 1 × . . . × ޚ p n r n , and so we have the following formula, where φ denotes the Euler phi function: ⎧ ⎨ 1 φ ( k ) + n , if k is odd , χ ( G ( ޚ k )) = 2 n ⎩ 2 , if k is even . Therefore, for example, the chromatic number of G ( ޚ 9 ) is equal to four and the chromatic number of G ( ޚ 12 ) is equal to two. In Figure 3, we illustrate these points. Here the different bullets indicate the presence of the different colours. E XAMPLE 3.10. The chromatic number of G ( ޚ 3 [ x ] / x 2 ) is equal to four. In Figure 4, we illustrate this point. Here the different bullets indicate the presence of the different colours. Note that this formula reduced to the simpler form if all of the R i ’s are fields. ⎧ n χ ( G ( R )) = ⎪ ⎨ 2 1 n ( | R i | − 1) + n , if 2 ∈ U ( R ) , ⎪ ⎩ i = 1 | R 1 | , if 2 ∈ / U ( R ) . We conclude this paper with two examples which illustrate the main result and the above formulas. E XAMPLE 3.9. Let k > 1 be an integer and write k = p r 1 1 . . . p r n n , where the p i ’s are distinct prime numbers and the r i ’s are positive integers. Therefore, we obtain ޚ k ∼ = ޚ p r 1 1 × . . . × ޚ p r n n , and so we have the following formula, where φ denotes the Euler phi function: ⎧ ⎨ 1 φ ( k ) + n , if k is odd , χ ( G ( ޚ k )) = 2 n ⎩ 2 , if k is even . Therefore, for example, the chromatic number of G ( ޚ 9 ) is equal to four and the chromatic number of G ( ޚ 12 ) is equal to two. In Figure 3, we illustrate these points. Here the different bullets indicate the presence of the different colours. E XAMPLE 3.10. The chromatic number of G ( ޚ 3 [ x ] / x 2 ) is equal to four. In Figure 4, we illustrate this point. Here the different bullets indicate the presence of the different colours. A CKNOWLEDGMENTS . The authors would like to thank the referee for his/her interest in the subject and for carefully reading the paper. The research of the authors was in part supported by a grant from IPM (Grant No. 88050214, 88130113 and ...
Context 2
... us note that the proof of Theorem 2.2 gives us an explicit formula for the chromatic number of G ( R ). For a given ring R , we may write R ∼ R × . . . × R , where Note that every this R i formula is a local reduced ring with to the maximal simpler ideal form m if i . all If of 2 ∈ / the U ( R R i ), ’s are by reordering, fields. we may assume that R 1 , . . . , R all ⎧ have characteristic equal to 2 with | R 1 | / | m 1 | ≤ . . . ≤ | R | / | m | and R + 1 , . . . , R n all have characteristic n not equal to 2. Then χ ( G ( R )) = ⎧ ⎨ ⎪ 2 1 n ( | R i | − 1) + n , if 2 ∈ U ( R ) , ⎩ ⎪ n i = 1 χ ( G ( R )) = ⎪ ⎨ 2 1 | n R 1 | , ( | R i | − | m i | ) + n , if if 2 2 ∈ / ∈ U U ( R ( R ) . ) , We conclude this paper with ⎪ ⎩ two i = 1 examples which illustrate the main result and the above formulas. | R | / | m | , if 2 ∈ / U ( R ) . E XAMPLE 3.9. Let k > 1 be an integer and write k = p r 1 1 . . . p r n n , where the p i ’s are distinct prime numbers and the r i ’s are positive integers. Therefore, we obtain ޚ k ∼ = ޚ p r 1 1 × . . . × ޚ p n r n , and so we have the following formula, where φ denotes the Euler phi function: ⎧ ⎨ 1 φ ( k ) + n , if k is odd , χ ( G ( ޚ k )) = 2 n ⎩ 2 , if k is even . Therefore, for example, the chromatic number of G ( ޚ 9 ) is equal to four and the chromatic number of G ( ޚ 12 ) is equal to two. In Figure 3, we illustrate these points. Here the different bullets indicate the presence of the different colours. E XAMPLE 3.10. The chromatic number of G ( ޚ 3 [ x ] / x 2 ) is equal to four. In Figure 4, we illustrate this point. Here the different bullets indicate the presence of the different colours. Note that this formula reduced to the simpler form if all of the R i ’s are fields. ⎧ n χ ( G ( R )) = ⎪ ⎨ 2 1 n ( | R i | − 1) + n , if 2 ∈ U ( R ) , ⎪ ⎩ i = 1 | R 1 | , if 2 ∈ / U ( R ) . We conclude this paper with two examples which illustrate the main result and the above formulas. E XAMPLE 3.9. Let k > 1 be an integer and write k = p r 1 1 . . . p r n n , where the p i ’s are distinct prime numbers and the r i ’s are positive integers. Therefore, we obtain ޚ k ∼ = ޚ p r 1 1 × . . . × ޚ p r n n , and so we have the following formula, where φ denotes the Euler phi function: ⎧ ⎨ 1 φ ( k ) + n , if k is odd , χ ( G ( ޚ k )) = 2 n ⎩ 2 , if k is even . Therefore, for example, the chromatic number of G ( ޚ 9 ) is equal to four and the chromatic number of G ( ޚ 12 ) is equal to two. In Figure 3, we illustrate these points. Here the different bullets indicate the presence of the different colours. E XAMPLE 3.10. The chromatic number of G ( ޚ 3 [ x ] / x 2 ) is equal to four. In Figure 4, we illustrate this point. Here the different bullets indicate the presence of the different colours. A CKNOWLEDGMENTS . The authors would like to thank the referee for his/her interest in the subject and for carefully reading the paper. The research of the authors was in part supported by a grant from IPM (Grant No. 88050214, 88130113 and ...
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Citations
... Recently, Ashrafi et al. [3] generalized the unit graph G(Z n ) to G(R) for an arbitrary ring R and obtained various results of finite commutative rings regarding the connectedness, the chromatic index, the diameter, the girth and the planarity of G(R). In recent years, many fundamental papers on unit graphs associated with rings have been appeared, for instance, see [14,17,16,15,19,20]. Nowadays, the study of graph structures on semiring theoretical setting is also an interesting area of research. ...
Let S be a commutative semiring with unity and U(S)
be the set of all units of S. The unit graph of S, denoted by
G(S) is the undirected graph with vertex set S and two distinct
vertices x and y are adjacent in G(S) if and only if x + y ∈ U(S).
In this paper, we concentrate on the unit graph G(S) and look
at several properties like the completeness, the bipartiteness, the
connectedness, the diameter and the girth. We also obtain
necessary and sufficient conditions for G(S) to be traversable under certain conditions.
... In addition to this, characterisations of finite commutative rings based on their diameter, girth, and planarity were also obtained in [225]. Using this structure of cliques and co-cliques, as well as the structural realisations obtained in in [225], the unit graph of a finite commutative ring was proven to be weakly perfect in [227], that is, for a finite commutative ring R, χ(G + (R)) = ω(G + (R)), where χ and ω denote the chromatic and the clique numbers of the graph. This was proven by using a series of lemmas, where finite commutative rings having different algebraic properties were considered, and the corresponding unit graphs were proven to be weakly perfect by computing their clique and chromatic numbers. ...
The study on graphs emerging from different algebraic structures such as groups, rings, fields, vector spaces, etc. is a prominent area of research in mathematics, as algebra and graph theory are two mathematical fields that focus on creating and analysing structures. There are numerous studies linking algebraic structures and graphs, which began with the introduction of Cayley graphs of groups. Several algebraic graphs have been defined on rings, a fast-growing area in the literature. In this article, we systematically review the literature on some variants of Cayley graphs that are defined on rings and highlight the properties and characteristics of such graphs, to showcase the research in this area.
... In addition to it, characterisations of finite commutative rings based on their diameter, girth and planarity were also obtained in [220]. Using this structure of cliques and co-cliques and the structural realisations obtained in in [220], the unit graph of a finite commutative ring was proved to be weakly perfect in [222]; that is, for a finite commutative ring R, χ(G + (R)) = ω(G + (R)), where χ and ω denote the chromatic and the clique number of the graph. This was proved by using a series of lemmas, where finite commutative rings having different algebraic properties were considered and the corresponding unit graphs were proved to be weakly perfect by computing their clique and chromatic numbers. ...
The study on graphs emerging from different algebraic structures like groups, rings, fields, vector spaces, etc. is a prominent area of research in mathematics, as algebra and graph theory are two mathematical fields that focuses on creating and analysing structures. There are numerous studies linking algebraic structures and graphs, which began with the introduction of Cayley graphs of groups. Several algebraic graphs have been defined on rings, which have huge-growing literature. In this article, we systematically review the literature on some variants of Cayley graphs that are defined on rings, to understand the research in this area.
... In 2010, Ashrafi et al. [8] generalized the unit graph G(Z n ) to G(R) for an arbitrary ring R. Many other papers are also devoted to this topic (see [9,36,37,38,39]). A survey of the study of unit graphs can be found in [35]. ...
This article is a survey of results concerning the eigenvalues of graphs constructed from commutative rings. Specifically, we consider the zero-divisor graphs, unitary Cayley graphs, unit graphs, and total graphs.
... In 1990, Grimaldi [16] discovered the unit graph for Z n : For an arbitrary ring R, Ashrafi et al. [7] generalized the unit graph GðZ n Þ to G(R) in 2010. There are numerous other articles on this specific topic (see [10,[23][24][25][26]). In [22], one can find a survey of the study on unit graphs. ...
This article presents a survey of results consisting of the Wiener index of graphs associated with commutative rings. In particular, we focus on zero-divisor graphs, unit graphs, total graphs and prime graphs. Further, we have posed some open problems corresponding to the graphs considered.
... In 1988, Beck [3] indroduced concept of the zero divisor graph Γ 0 (R) of R, where V (Γ 0 (R)) = R, and two distinct vertices x and y of R are adjacent if and only if xy = 0. This investigation of graphs associated with rings has been continued by many mathematicians, for instance see [1,2,5,7,9,11,12]. Anderson and Badawi [1] introduced and investigated the total graph of finite ring, denoted by T (Γ(R)). ...
... Ashrafi et al. [2], generalized the unit graph G(Z n ) of ring Z n to unit graph G(R) of R, where R is an arbitrary ring. After that, many mathematicians studied about the various graph theoretical aspects of the unit graphs of commutative rings in [9,11,12]. In this paper, we introduce a new unit graph structures associated with a finite ring Z m × Z n , where m, n > 1 be any two integers, and we denote the unit graph of Z m × Z n by G(Z m × Z n ). ...
... 2 We recall the following result to investigate the chromatic number of G(Z m × Z n ). Theorem 26 [12,Theorem 2.2]. If R is a ring, then the unit graph G(R) is weakly perfect i.e. ω(G(R)) = χ(G(R)). ...
Let R be a commutative ring with unity and U(R) be the set of all units of R. The unit graph of R, denoted by G(R), is the (undirected) graph with vertex set R and two distinct vertices x and y in G(R) are adjacent if and only if x + y ∈ U(R). In this paper, we investigate certain fundamental properties of the unit graph of Zm × Zn, where m, n > 1 be any two integers. We also give the traversability properties of the unit graph of Zm × Zn. Further, we determine the clique number, the chromatic number, the independent number, and the covering number of the unit graph of Zm × Zn.
... Definition 2.2. [12] Let R be a ring and U (R) be the set of unit elements of R. The unit graph of R, denoted by U (R), is the graph obtained by setting all the elements of R to be the vertices and defining distinct vertices x and y to be adjacent if and only if x + y ∈ U (R). Figure 1: The unit graphs of two isomorphic rings Z 3 × Z 2 and Z 6 Definition 2.3. [14] The coprime graph Γ G with a finite G is defined as follows: Take V (G) as the vertex set of Γ G and join two distinct vertices x and y if (o|x|, o|y|) = 1. ...
Given a distribution on the vertices of a connected graph, a pebbling move on G consists of removing two pebbles from one vertex, placing one on an adjacent vertex and throwing away another pebble. The pebbling number, f (G), of a connected graph G, is the smallest positive integer such that any distribution of f (G) pebbles on G allows one pebble to be moved to any specified vertex by a sequence of pebbling moves. In this paper, we compute the pebbling number for the Unit graphs and Co-prime graphs.
... Definition 2.2. [12] Let R be a ring and U (R) be the set of unit elements of R. The unit graph of R, denoted by G(R), is the graph obtained by setting all the elements of R to be the vertices and defining distinct vertices x and y to be adjacent if and only if x + y ∈ U (R). The coprime graph Γ G with a finite group is defined as G as follows: Take G as the vertex set of Γ G and join two distinct vertices ...
Given a distribution on the vertices of a connected graph, a pebbling move
on G consists of removing two pebbles from one vertex, placing one on an
adjacent vertex and throwing away another pebble. The pebbling number,
f(G), of a connected graph G, is the smallest positive integer such that any
distribution of f(G) pebbles on G allows one pebble to be moved to any
specified vertex by a sequence of pebbling moves. In this paper, we compute
the pebbling number for the Unit graphs and Co-prime graphs.
... In 1990, the unit graph was first investigated by Chung [13] and Grimaldi [16] for Z n . In 2010, Ashrafi, et al. [7] generalized the unit graph G(Z n ) to G(R) for an arbitrary ring R. Numerous results about unit and unitary Cayley graphs were obtained, see for examples [3,7,18,19,21,22]. ...
Let \({E}_{n}\) be the ring of Eisenstein integers modulo \(n\). We denote by \(G({E}_{n})\) and \(G_{{E}_{n}}\), the unit graph and the unitary Cayley graph of \({E}_{n}\), respectively. In this paper, we obtain the value of the diameter, the girth, the clique number and the chromatic number of these graphs. We also prove that for each \(n>1\), the graphs \(G(E_{n})\) and \(G_{E_{n}}\) are Hamiltonian.
... Ashrafi et al. [2] generalized unit graph of Z n to an arbitrary ring R and studied the properties of this graph. Results in [9] were generalized to a finite commutative ring by the same author in [10]. Indeed, they proved that unit graph of a finite commutative ring is also weakly perfect. ...
... The unit graph is also the topic of several other publications (see for example [1], [5], [8], [12] and [13]). The main aim of this paper is to strengthen results in [9,10]; In other word, we study perfect unit graphs of Artinin rings (not necessarily finite). Let us first recall some necessary definitions and notations. ...
Let R be a commutative ring with identity. The unit graph of R, denoted by G(R), has its set of vertices equal to the set of all elements of R and two distinct vertices x and y are adjacent if and only if \(x+y\) is a unit of R. In this paper, perfect unit graphs of Artinian rings are investigated.
