Eigenvalue Characterizations for the Signless Laplacian Spectrum of Weakly Zero-Divisor Graphs on Z n

Nazim (), Alaa Altassan and Nof T. Alharbi
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Nazim: Department of Applied Science, Meerut Institue of Engineering and Technology, Meerut 250005, India
Alaa Altassan: Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Nof T. Alharbi: Mathematics Department, University College in Al-Darb, Jazan University, Jazan 82817, Saudi Arabia

Mathematics, 2025, vol. 13, issue 16, 1-19

Abstract: Let R be a commutative ring with identity 1 ≠ 0 . The weakly zero-divisor graph of R , denoted W Γ ( R ) , is the simple undirected graph whose vertex set consists of the nonzero zero-divisors of R , where two distinct vertices a and b are adjacent if and only if there exist r ∈ ann ( a ) and s ∈ ann ( b ) such that r s = 0 . In this paper, we study the signless Laplacian spectrum of W Γ ( Z n ) for several composite forms of n , including n = p 2 q 2 , n = p 2 q r , n = p m q m and n = p m q r , where p , q , r are distinct primes and m ≥ 2 . By using generalized join decomposition and quotient matrix methods, we obtain explicit eigenvalue formulas for each case, along with structural bounds, spectral integrality conditions and Nordhaus–Gaddum-type inequalities. Illustrative examples with computed spectra are provided to validate the theoretical results, demonstrating the interplay between the algebraic structure of Z n and the spectral properties of its weakly zero-divisor graph.

Keywords: weakly zero-divisor graph; signless Laplacian spectrum; finite commutative ring; Euler totient function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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