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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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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author Amal Alsaluli
Wafaa Fakieh
Hanaa Alashwali
author_facet Amal Alsaluli
Wafaa Fakieh
Hanaa Alashwali
author_sort Amal Alsaluli
collection DOAJ
description 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>.
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spelling doaj-art-87988fd405c1477d8fbb0cea2f38cabd2025-08-20T02:43:28ZengMDPI AGAxioms2075-16802024-12-01131287310.3390/axioms13120873Laplacian Spectrum and Vertex Connectivity of the Unit Graph of the Ring \({ℤ_{p^{r}q^{s}}}\)Amal Alsaluli0Wafaa Fakieh1Hanaa Alashwali2Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi ArabiaDepartment of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi ArabiaDepartment of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi ArabiaIn 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>.https://www.mdpi.com/2075-1680/13/12/873unit graph<i>G</i>-generalized join graphLaplacian spectral radiusalgebraic connectivityring of integers modulo <i>n</i>
spellingShingle Amal Alsaluli
Wafaa Fakieh
Hanaa Alashwali
Laplacian Spectrum and Vertex Connectivity of the Unit Graph of the Ring \({ℤ_{p^{r}q^{s}}}\)
Axioms
unit graph
<i>G</i>-generalized join graph
Laplacian spectral radius
algebraic connectivity
ring of integers modulo <i>n</i>
title Laplacian Spectrum and Vertex Connectivity of the Unit Graph of the Ring \({ℤ_{p^{r}q^{s}}}\)
title_full Laplacian Spectrum and Vertex Connectivity of the Unit Graph of the Ring \({ℤ_{p^{r}q^{s}}}\)
title_fullStr Laplacian Spectrum and Vertex Connectivity of the Unit Graph of the Ring \({ℤ_{p^{r}q^{s}}}\)
title_full_unstemmed Laplacian Spectrum and Vertex Connectivity of the Unit Graph of the Ring \({ℤ_{p^{r}q^{s}}}\)
title_short Laplacian Spectrum and Vertex Connectivity of the Unit Graph of the Ring \({ℤ_{p^{r}q^{s}}}\)
title_sort laplacian spectrum and vertex connectivity of the unit graph of the ring z p r q s
topic unit graph
<i>G</i>-generalized join graph
Laplacian spectral radius
algebraic connectivity
ring of integers modulo <i>n</i>
url https://www.mdpi.com/2075-1680/13/12/873
work_keys_str_mv AT amalalsaluli laplacianspectrumandvertexconnectivityoftheunitgraphoftheringzprqs
AT wafaafakieh laplacianspectrumandvertexconnectivityoftheunitgraphoftheringzprqs
AT hanaaalashwali laplacianspectrumandvertexconnectivityoftheunitgraphoftheringzprqs

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