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• W4225829502 abstract "Abstract 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> Gamma left({{mathbb{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> {{mathbb{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> ngt 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> Gamma left({{mathbb{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> Gamma left({{mathbb{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> Gamma left({{mathbb{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> Gamma left({{mathbb{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}^{alpha }{q}^{beta } , 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 are primes and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> </m:math> alpha ,beta are non-negative integers and hence calculate the number of spanning trees 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> Gamma left({{mathbb{Z}}}_{n}) 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}^{alpha }{q}^{beta } . The algebraic and vertex connectivity 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> Gamma left({{mathbb{Z}}}_{n}) have been shown to be equal for all <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>n</m:mi> </m:math> n . An upper bound on the second largest Laplacian eigenvalue 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> Gamma left({{mathbb{Z}}}_{n}) has been obtained, and a necessary and sufficient condition for its equality has also been determined. Finally, we discuss the multiplicity of the Laplacian spectral radius and the multiplicity of the algebraic connectivity 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> Gamma left({{mathbb{Z}}}_{n}) . We then investigate some properties and vertex connectivity of an induced subgraph 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> Gamma left({{mathbb{Z}}}_{n}) . Some problems have been discussed at the end of this paper for further research." @default.
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• W4225829502 date "2022-01-01" @default.
• W4225829502 modified "2023-10-14" @default.
• W4225829502 title "Laplacian spectrum of comaximal graph of the ring ℤ_<i>n</i>" @default.
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• W4225829502 doi "https://doi.org/10.1515/spma-2022-0163" @default.
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