| Summary: | Given a commutative ring <i>R</i> with identity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula>, let the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> denote the set of zero-divisors and let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>Z</mi><mo>*</mo></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>=</mo><mi>Z</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>∖</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula> be the set of non-zero zero-divisors of <i>R</i>. The zero-divisor graph of <i>R</i>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Γ</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula>, is a simple graph whose vertex set is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>Z</mi><mo>*</mo></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and each pair of vertices in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>Z</mi><mo>*</mo></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are adjacent when their product is 0. In this article, we find the structure and Laplacian spectrum of the zero-divisor graphs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Γ</mi><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><msup><mi>p</mi><msub><mi>N</mi><mn>1</mn></msub></msup><msup><mi>q</mi><msub><mi>N</mi><mn>2</mn></msub></msup></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo><</mo><mi>q</mi></mrow></semantics></math></inline-formula> are primes and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>N</mi><mn>1</mn></msub><mo>,</mo><msub><mi>N</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> are positive integers.
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