A note on eigenvalues of zero divisor graphs associated with commutative rings

2024/01/04 by Bilal Ahmad Rather, Rather, Bilal Ahmad
Materials Science · Mathematics · Physics and Astronomy · #05C25 #05C50 #13A70 #15A18 #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Graph theory and applications #Magnetism in coordination complexes #Quantum and electron transport phenomena #Rings and Algebras (math.RA) #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2401.02554

openalex publication_date 2024/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a commutative ring R, with non-zero zero divisors Z^∗(R). The zero divisor graph Γ(R) is a simple graph with vertex set Z^∗(R), and two distinct vertices x,y∈ V(Γ(R)) are adjacent if and only if x⋅ y=0. In this note, we provide counter examples to the eigenvalues, the energy and the second Zagreb index related to zero divisor graphs of rings obtained in [Johnson and Sankar, J. Appl. Math. Comp. (2023), \citejohnson]. We correct the eigenvalues (energy) and the Zagreb index result for the zero divisor graphs of ring ℤ_p[x]/⟨ x⁴ ⟩. We show that for any prime p, Γ(ℤ_p[x]/⟨ x⁴ ⟩) is non-hyperenergetic and for prime p≥ 3, Γ(ℤ_p[x]/⟨ x⁴ ⟩) is hypoenergetic. We give a formulae for the topological indices of Γ(ℤ_p[x]/⟨ x⁴ ⟩) and show that its Zagreb indices satisfy Hansen and Vuki\checkccević conjecture \citehansen.

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