Common Neighborhood Energy of the Non-Commuting Graphs and Commuting Graphs Associated with Dihedral and Generalized Quaternion Groups
This paper explores the common neighborhood energy (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>E</mi><mrow><mi>C</mi><mi>N</mi></mrow>&l...
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2025-05-01
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| author | Hanaa Alashwali Anwar Saleh |
| author_facet | Hanaa Alashwali Anwar Saleh |
| author_sort | Hanaa Alashwali |
| collection | DOAJ |
| description | This paper explores the common neighborhood energy (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>E</mi><mrow><mi>C</mi><mi>N</mi></mrow></msub><mrow><mo>(</mo><mi mathvariant="sans-serif">Γ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>) of graphs derived from the dihedral group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub></semantics></math></inline-formula> and generalized quaternion group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Q</mi><mrow><mn>4</mn><mi>n</mi></mrow></msub></semantics></math></inline-formula>, specifically the non-commuting graph (NCM-graph) and the commuting graph (CM-graph). Studying graphs associated with groups offers a powerful approach to translating algebraic properties into combinatorial structures, enabling the application of graph-theoretic tools to understand group behavior. The common neighborhood energy, defined as the sum of the absolute values of the eigenvalues of the common neighborhood (CN) matrix, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></msubsup><mrow><mo>|</mo><msub><mi>ζ</mi><mi>i</mi></msub><mo>|</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo>{</mo><msub><mi>ζ</mi><mi>i</mi></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></msubsup></semantics></math></inline-formula> are the CN eigenvalues, provides insights into the structural properties of these graphs. We derive explicit formulas for the CN characteristic polynomials and corresponding CN eigenvalues for both the NCM-graph and CM-graph as functions of <i>n</i>. Consequently, we establish closed-form expressions for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mrow><mi>C</mi><mi>N</mi></mrow></msub></semantics></math></inline-formula> of these graphs, which are parameterized by <i>n</i>. The validity of our theoretical results is confirmed through computational examples. This study contributes to the spectral analysis of algebraic graphs, demonstrating a direct connection between the group-theoretic structure of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Q</mi><mrow><mn>4</mn><mi>n</mi></mrow></msub></semantics></math></inline-formula>, as well as the combinatorial energy of their associated graphs, thus furthering the understanding of group properties through spectral graph theory. |
| format | Article |
| id | doaj-art-46fb7b608cf74e37b34605b611c41dbb |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-46fb7b608cf74e37b34605b611c41dbb2025-08-20T03:11:32ZengMDPI AGMathematics2227-73902025-05-011311183410.3390/math13111834Common Neighborhood Energy of the Non-Commuting Graphs and Commuting Graphs Associated with Dihedral and Generalized Quaternion GroupsHanaa Alashwali0Anwar Saleh1Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi ArabiaDepartment of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23218, Saudi ArabiaThis paper explores the common neighborhood energy (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>E</mi><mrow><mi>C</mi><mi>N</mi></mrow></msub><mrow><mo>(</mo><mi mathvariant="sans-serif">Γ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>) of graphs derived from the dihedral group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub></semantics></math></inline-formula> and generalized quaternion group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Q</mi><mrow><mn>4</mn><mi>n</mi></mrow></msub></semantics></math></inline-formula>, specifically the non-commuting graph (NCM-graph) and the commuting graph (CM-graph). Studying graphs associated with groups offers a powerful approach to translating algebraic properties into combinatorial structures, enabling the application of graph-theoretic tools to understand group behavior. The common neighborhood energy, defined as the sum of the absolute values of the eigenvalues of the common neighborhood (CN) matrix, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></msubsup><mrow><mo>|</mo><msub><mi>ζ</mi><mi>i</mi></msub><mo>|</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo>{</mo><msub><mi>ζ</mi><mi>i</mi></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></msubsup></semantics></math></inline-formula> are the CN eigenvalues, provides insights into the structural properties of these graphs. We derive explicit formulas for the CN characteristic polynomials and corresponding CN eigenvalues for both the NCM-graph and CM-graph as functions of <i>n</i>. Consequently, we establish closed-form expressions for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mrow><mi>C</mi><mi>N</mi></mrow></msub></semantics></math></inline-formula> of these graphs, which are parameterized by <i>n</i>. The validity of our theoretical results is confirmed through computational examples. This study contributes to the spectral analysis of algebraic graphs, demonstrating a direct connection between the group-theoretic structure of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Q</mi><mrow><mn>4</mn><mi>n</mi></mrow></msub></semantics></math></inline-formula>, as well as the combinatorial energy of their associated graphs, thus furthering the understanding of group properties through spectral graph theory.https://www.mdpi.com/2227-7390/13/11/1834CN energyNCM-graphCM-graphdihedral groupsgeneralized quaternion groups |
| spellingShingle | Hanaa Alashwali Anwar Saleh Common Neighborhood Energy of the Non-Commuting Graphs and Commuting Graphs Associated with Dihedral and Generalized Quaternion Groups Mathematics CN energy NCM-graph CM-graph dihedral groups generalized quaternion groups |
| title | Common Neighborhood Energy of the Non-Commuting Graphs and Commuting Graphs Associated with Dihedral and Generalized Quaternion Groups |
| title_full | Common Neighborhood Energy of the Non-Commuting Graphs and Commuting Graphs Associated with Dihedral and Generalized Quaternion Groups |
| title_fullStr | Common Neighborhood Energy of the Non-Commuting Graphs and Commuting Graphs Associated with Dihedral and Generalized Quaternion Groups |
| title_full_unstemmed | Common Neighborhood Energy of the Non-Commuting Graphs and Commuting Graphs Associated with Dihedral and Generalized Quaternion Groups |
| title_short | Common Neighborhood Energy of the Non-Commuting Graphs and Commuting Graphs Associated with Dihedral and Generalized Quaternion Groups |
| title_sort | common neighborhood energy of the non commuting graphs and commuting graphs associated with dihedral and generalized quaternion groups |
| topic | CN energy NCM-graph CM-graph dihedral groups generalized quaternion groups |
| url | https://www.mdpi.com/2227-7390/13/11/1834 |
| work_keys_str_mv | AT hanaaalashwali commonneighborhoodenergyofthenoncommutinggraphsandcommutinggraphsassociatedwithdihedralandgeneralizedquaterniongroups AT anwarsaleh commonneighborhoodenergyofthenoncommutinggraphsandcommutinggraphsassociatedwithdihedralandgeneralizedquaterniongroups |