Eigenvalues of zero-divisor graphs of finite commutative rings

Mönius, Katja

doi:10.1007/s10801-020-00989-6

Cited by 16 publications

(3 citation statements)

References 12 publications

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“…Moreover, the realizations of rings as graphs were introduced in [1,3]. The aim of considering these realizations of rings as graphs is to study the interplay between combinatorial and ring theoretic properties of a ring R. This concept was further studied in [16,[18][19][20] and was extended to modules over commutative rings in [21].…”

Section: The Above Combinatorial Description Is One Of the Exploratio...mentioning

confidence: 99%

Degenerations and order of graphs realized by finite abelian groups

Raja¹

2022

Preprint

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Let G 1 and G 2 be two groups. If there exists a homomorphism ϕ : G 1 → G 2 such that ϕ(a) = b, where a ∈ G 1 and b ∈ G 2 , then a group G 1 degenerates to a group G 2 . In this paper, we study degeneration in graphs and show that degeneration in groups is a particular case of degeneration in graphs. We exhibit some interesting properties of degeneration in graphs. We use this concept to present a pictorial representation of graphs realized by finite abelian groups. We discus some partial orders on the set T p 1 •••p n of all graphs realized by finite abelian p r -groups, where each p r , 1 ≤ r ≤ n, is a prime number. We show that each finite abelian p r -group of rank n can be identified with saturated chains of Young diagrams in the poset T p 1 •••p n . We present a combinatorial formula which represents the degree of a projective representation of a symmetric group. This formula determines the number of different saturated chains in T p 1 •••p n and the number of finite abelian groups of different orders.

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Section: The Above Combinatorial Description Is One Of the Exploratio...mentioning

confidence: 99%

Degenerations and order of graphs realized by finite abelian groups

Raja¹

2022

Preprint

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“…The zero-divisor graph of a commutative ring R with unity was introduced by Beck [5]. The graph Γ(R) is defined to be the graph with vertex set as R and two distinct vertices x, y ∈ R are adjacent (that is, x ∼ y) in Γ(R) if and only if xy = 0 in R. The notion of zero-divisor graphs was further studied by Anderson and Livingston in [1], who restricted vertices of the graph Γ(R) to non-zero zero-divisors of ring R. The main objective of the investigation of associating a graph to R is to study the interplay of graph theoretic properties of Γ(R) and ring theoretic properties of R. This interplay between different properties of R and Γ(R) was thoroughly investigated in [2,4,15,17,18,19].…”

Section: Introductionmentioning

confidence: 99%

Realization of zero-divisor graphs of finite commutative rings as threshold graphs

Raja¹,

Wagay²

2022

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Let R be a finite commutative ring with unity and let G = (V, E) be a simple graph. The zero-divisor graph denoted by Γ(R) is a simple graph with vertex set as R and two vertices x, y ∈ R are adjacent in Γ(R) if and only if xy = 0. In [6], the authors have studied the Laplacian eigen values of the graph Γ(Zn) and for distinct proper divisors d1, d2, • • • , d k of n, they defined the sets aswhere (x, n) denotes the greatest common divisor of x and n. In this paper, we show that the sets A d i , where 1 ≤ i ≤ k are actually the orbits of the group action: Aut(Γ(R)) × R −→ R, where Aut(Γ(R)) denotes the automorphism group of Γ(R). We characterize all finite commutative rings with unity of which zero-divisor graphs are not threshold. We study creation sequences, hyperenergeticity and hypoenergeticity of zero-divisor graphs. We compute the Laplacian eigenvalues of zero-divisor graphs realized by some classes of reduced and local rings. We show that the Laplacian eigenvalues of zero-divisor graphs are the representatives of orbits of the group action:Aut(Γ(R)) × R −→ R.

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“…Anderson and Livingston [1] also investigated the combinatorial properties of a commutative ring R. They associated a graph Γ(R) to R with vertices as elements of Z * (R) = Z(R) \ {0}, that is, the non-zero zero-divisors of R with two vertices a, b ∈ Z * (R) are adjacent in Γ(R) if and only if ab = 0. For more on zero-divisor graphs please see [2,13,14,15,16].…”

Section: Introductionmentioning

confidence: 99%

Lambda number of zero-divisor graphs of finite commutative rings

Ali¹,

Raja²

2022

Preprint

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Let G = (V, E) be a simple graph, an L(2, 1)-labelling of G is assignment of labels from non-negative integers to the vertices of G such that adjacent vertices gets labels which at least differ by two and vertices which are at distance two from each other get different labels. The λ-number of G, denoted by λ(G) is the smallest positive integer such that G has a L(2, 1)labelling with all the labels are members of the set {0, 1, • • • , }. The zero-divisor graph denoted by Γ(R), of a finite commutative ring R with unity is a simple graph with vertices as non-zero zero divisors of R. Two vertices u and v are adjacent in Γ(R) if and only if uv = 0 in R. In this paper, we investigate L(2, 1)-labelling in zero-divisor graphs. We study the partite truncation, a graph operation that reduces a n-partite graph of higher order to a graph of lower order. We establish the relation between λ-numbers of two graphs. We make use of the operation partite truncation to contract the zero-divisor graph of a reduced ring to the zero-divisor graph of a Boolean ring. We compute the exact value of λ-numbers of zero-divisor graphs of some classes of local and mixed rings such as Z p n , Z p n × Z q m , and F q × Z p n .

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Eigenvalues of zero-divisor graphs of finite commutative rings

Abstract: We investigate eigenvalues of the zero-divisor graph $$\Gamma (R)$$ Γ ( R ) of finite commutative rings R and study the interplay between these eigenvalues, the ring-theoretic properties of R and the graph-theoretic properties of $$\Gamma (R)$$ Γ ( R ) . The graph $$\Gamma … Show more

Cited by 16 publications

References 12 publications

Degenerations and order of graphs realized by finite abelian groups

Degenerations and order of graphs realized by finite abelian groups

Realization of zero-divisor graphs of finite commutative rings as threshold graphs

Lambda number of zero-divisor graphs of finite commutative rings

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