Eigenvalue Characterizations for the Signless Laplacian Spectrum of Weakly Zero-Divisor Graphs on <inline-formula><math display="inline"><semantics><mrow><msub style="bold"><mi mathvariant="double-struck">Z</mi><mi mathvariant="italic">n</mi></msub></mrow></semantics></math></inline-formula>

Abstract

Let R be a commutative ring with identity 1≠0. The weakly zero-divisor graph of R, denoted WΓ(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 rs=0. In this paper, we study the signless Laplacian spectrum of WΓ(Zn) for several composite forms of n, including n=p2q2, n=p2qr, n=pmqm and n=pmqr, 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 Zn and the spectral properties of its weakly zero-divisor graph.

Keywords

• weakly zero-divisor graph
• signless Laplacian spectrum
• finite commutative ring
• Euler totient function