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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-05-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/11/1834 |
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| Summary: | 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. |
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| ISSN: | 2227-7390 |