Synchronization Problems Solvable by Generalized PV Systems

Henderson, Peter B.; Zalcstein, Yechezkel

doi:10.1145/322169.322175

Cited by 11 publications

(1 citation statement)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The family of threshold graphs represent an important class of simple graphs. The concept of threshold graphs was first established by the authors Chvátal and Hammer in [7] and Henderson and Zalcstein in [10].…”

Section: Introductionmentioning

confidence: 99%

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

Raja¹,

Wagay²

2022

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%