Laplacian spectrum of comaximal graph of the ring ℤ n

In this paper, we study the interplay between the structural and spectral properties of the comaximal graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Γ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">Z</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> Γ ({{Z}}ₙ) of the ring <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">Z</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> {{Z}}ₙ for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> <m:mo>></m:mo> <m:mn>2</m:mn> </m:math> n> 2 . We first determine the structure of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Γ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">Z</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> Γ ({{Z}}ₙ) and deduce some of its properties. We then use the structure of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Γ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">Z</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> Γ ({{Z}}ₙ) to deduce the Laplacian eigenvalues of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Γ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">Z</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> Γ ({{Z}}ₙ) for various <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> </m:math> n . We show that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Γ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">Z</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> Γ ({{Z}}ₙ) is Laplacian integral for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mi>β</m:mi> </m:mrow> </m:msup> </m:math> n={p}α{q}β , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:math> p,q </jats: