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

Rameez Raja () and Samir Ahmad Wagay ()
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Rameez Raja: National Institute of Technology Srinagar
Samir Ahmad Wagay: National Institute of Technology Srinagar

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 2, 567-576

Abstract: Abstract Let R be a finite commutative ring with unity, and let $$G = (V, E)$$ G = ( V , E ) be a simple graph. The zero-divisor graph, denoted by $$\Gamma (R)$$ Γ ( R ) is a simple graph with vertex set as R, and two vertices $$x, y \in R$$ x , y ∈ R are adjacent in $$\Gamma (R)$$ Γ ( R ) if and only if $$xy=0$$ x y = 0 . In [5], the authors have studied the Laplacian eigenvalues of the graph $$\Gamma (\mathbb {Z}_{n})$$ Γ ( Z n ) and for distinct proper divisors $$d_1, d_2, \cdots , d_k$$ d 1 , d 2 , ⋯ , d k of n, they defined the sets as, $$\mathcal {A}_{d_i} = \{x \in \mathbb {Z}_{n} : (x, n) = d_i\}$$ A d i = { x ∈ Z n : ( x , n ) = d i } , where (x, n) denotes the greatest common divisor of x and n. In this paper, we show that the sets $$\mathcal {A}_{d_i}$$ A d i , $$1 \le i \le k$$ 1 ≤ i ≤ k are actually orbits of the group action: $$Aut(\Gamma (R)) \times R \longrightarrow R$$ A u t ( Γ ( R ) ) × R ⟶ R , where $$Aut(\Gamma (R))$$ A u t ( Γ ( R ) ) denotes the automorphism group of $$\Gamma (R)$$ Γ ( R ) . Our main objective is to determine new classes of threshold graphs, since these graphs play an important role in several applied areas. For a reduced ring R, we prove that $$\Gamma (R)$$ Γ ( R ) is a connected threshold graph if and only if $$R\cong {F}_{q}$$ R ≅ F q or $$R\cong {F}_2 \times {F}_{q}$$ R ≅ F 2 × F q . We provide classes of threshold graphs realized by some classes of local rings. Finally, we characterize all finite commutative rings with unity of which zero-divisor graphs are not threshold.

Keywords: Group action; Orbits; Zero-divisor; Zero-divisor graph; Threshold graph; 13A70; 05C25; 05C50 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13226-023-00389-z

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