Laplacian Spectrum and Vertex Connectivity of the Unit Graph of the Ring \({ℤ_{p^{r}q^{s}}}\)

Laplacian Spectrum and Vertex Connectivity of the Unit Graph of the Ring \({ℤ_{p^{r}q^{s}}}\)

In this paper, we examine the interplay between the structural and spectral properties of the unit graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>(</mo><msub...

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Bibliographic Details
Main Authors: Amal Alsaluli, Wafaa Fakieh, Hanaa Alashwali
Format: Article
Language:English
Published: MDPI AG 2024-12-01
Series:Axioms
Subjects:
unit graph
<i>G</i>-generalized join graph
Laplacian spectral radius
algebraic connectivity
ring of integers modulo <i>n</i>
Online Access:https://www.mdpi.com/2075-1680/13/12/873
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Summary:In this paper, we examine the interplay between the structural and spectral properties of the unit graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</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><msubsup><mi>p</mi><mn>1</mn><msub><mi>r</mi><mn>1</mn></msub></msubsup><msubsup><mi>p</mi><mn>2</mn><msub><mi>r</mi><mn>2</mn></msub></msubsup><mo>…</mo><msubsup><mi>p</mi><mi>k</mi><msub><mi>r</mi><mi>k</mi></msub></msubsup></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>p</mi><mn>1</mn></msub><mo>,</mo><msub><mi>p</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>p</mi><mi>k</mi></msub></mrow></semantics></math></inline-formula> are distinct primes and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>,</mo><msub><mi>r</mi><mn>1</mn></msub><mo>,</mo><msub><mi>r</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>r</mi><mi>k</mi></msub></mrow></semantics></math></inline-formula> are positive integers such that at least one of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>r</mi><mi>i</mi></msub></semantics></math></inline-formula> must be greater than 1. We first analyze the structure of the unit graph of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mi>n</mi></msub></semantics></math></inline-formula>, treating it as what we will define as a ‘generalized join graph’ under these conditions. We then determine the Laplacian spectrum of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> and deduce that it is integral for all <i>n</i>. Consequently, we obtain the Laplacian spectral radius and algebraic connectivity of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>. We also prove that the vertex connectivity of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mrow><mi>p</mi><mi>q</mi></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula> is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>2</mn><mo>)</mo><mi>q</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>≠</mo><mi>p</mi><mo><</mo><mi>q</mi></mrow></semantics></math></inline-formula>. We deduce the vertex connectivity of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><msup><mi>p</mi><mi>r</mi></msup><msup><mi>q</mi><mi>s</mi></msup></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>≠</mo><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><mi>r</mi><mo>,</mo><mi>s</mi></mrow></semantics></math></inline-formula> are positive integers. Finally, we present conjectures regarding the vertex connectivity of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><msub><mi>p</mi><mn>1</mn></msub><msub><mi>p</mi><mn>2</mn></msub><mo>…</mo><msub><mi>p</mi><mi>k</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><msubsup><mi>p</mi><mn>1</mn><msub><mi>r</mi><mn>1</mn></msub></msubsup><msubsup><mi>p</mi><mn>2</mn><msub><mi>r</mi><mn>2</mn></msub></msubsup><mo>…</mo><msubsup><mi>p</mi><mi>k</mi><msub><mi>r</mi><mi>k</mi></msub></msubsup></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>p</mi><mi>i</mi></msub></semantics></math></inline-formula> are distinct primes, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>r</mi><mi>i</mi></msub></semantics></math></inline-formula> are positive integers, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></mrow></semantics></math></inline-formula>.
ISSN:2075-1680

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