Laplacian eigenvalues of the zero divisor graph of the ring\n \ℤ_n

2019/03/19 by Sriparna Chattopadhyay, Kamal Lochan Patra, Chattopadhyay, Sriparna +3
Chemistry · Mathematics · #05C25 #05C50 #05C75 #Advanced NMR Techniques and Applications #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Spectral Theory (math.SP) #Synthesis and Properties of Aromatic Compounds

paper · pdf · doi:10.48550/arxiv.1903.07841

openalex publication_date 2019/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Laplacian eigenvalues of the zero divisor graph\n\Γ\(\ℤ_n\) of the ring \ℤ_n and prove that\n\Γ\(\ℤ_p^t\) is Laplacian integral for every prime p\nand positive integer t\≥ 2. We also prove that the Laplacian spectral\nradius and the algebraic connectivity of \Γ\(\ℤ_n\)\nfor most of the values of n are, respectively, the largest and the second\nsmallest eigenvalues of the vertex weighted Laplacian matrix of a graph which\nis defined on the set of proper divisors of n. The values of n for which\nalgebraic connectivity and vertex connectivity of\n\Γ\(\ℤ_n\) coincide are also characterized.\n

Citations

• The Zero-Divisor Graph of a Commutative Ring
• Coloring of commutative rings
• On graphs with equal algebraic and vertex connectivity

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