On the Extended Adjacency Eigenvalues of a Graph
Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> be a graph of order <i>n</i> with <i>m&...
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2024-09-01
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| author | Alaa Altassan Hilal A. Ganie Yilun Shang |
| author_facet | Alaa Altassan Hilal A. Ganie Yilun Shang |
| author_sort | Alaa Altassan |
| collection | DOAJ |
| description | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> be a graph of order <i>n</i> with <i>m</i> edges. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi>i</mi></msub><mo>=</mo><mi>d</mi><mrow><mo>(</mo><msub><mi>v</mi><mi>i</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> be the degree of the vertex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>i</mi></msub></semantics></math></inline-formula>. The extended adjacency matrix <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mrow><mi>e</mi><mi>x</mi></mrow></msub><mrow><mo>(</mo><mi mathvariant="script">H</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> is an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula> matrix defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mrow><mi>e</mi><mi>x</mi></mrow></msub><mrow><mo>(</mo><mi mathvariant="script">H</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><msub><mi>b</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>b</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mfenced separators="" open="(" close=")"><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>d</mi><mi>i</mi></msub><msub><mi>d</mi><mi>j</mi></msub></mfrac></mstyle><mo>+</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>d</mi><mi>j</mi></msub><msub><mi>d</mi><mi>i</mi></msub></mfrac></mstyle></mfenced></mrow></semantics></math></inline-formula>, whenever <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>i</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>j</mi></msub></semantics></math></inline-formula> are adjacent and equal to zero otherwise. The largest eigenvalue of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mrow><mi>e</mi><mi>x</mi></mrow></msub><mrow><mo>(</mo><mi mathvariant="script">H</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is called the extended adjacency spectral radius of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> and the sum of the absolute values of its eigenvalues is called the extended adjacency energy of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula>. In this paper, we obtain some sharp upper and lower bounds for the extended adjacency spectral radius in terms of different graph parameters and characterize the extremal graphs attaining these bounds. We also obtain some new bounds for the extended adjacency energy of a graph and characterize the extremal graphs attaining these bounds. In both cases, we show our bounds are better than some already known bounds in the literature. |
| format | Article |
| id | doaj-art-de3449aaa2ec4176af18c3777568206f |
| institution | OA Journals |
| issn | 2078-2489 |
| language | English |
| publishDate | 2024-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Information |
| spelling | doaj-art-de3449aaa2ec4176af18c3777568206f2025-08-20T02:11:04ZengMDPI AGInformation2078-24892024-09-01151058610.3390/info15100586On the Extended Adjacency Eigenvalues of a GraphAlaa Altassan0Hilal A. Ganie1Yilun Shang2Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi ArabiaDepartment of Mathematics, University of Kashmir, Srinagar 190001, IndiaDepartment of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UKLet <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> be a graph of order <i>n</i> with <i>m</i> edges. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi>i</mi></msub><mo>=</mo><mi>d</mi><mrow><mo>(</mo><msub><mi>v</mi><mi>i</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> be the degree of the vertex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>i</mi></msub></semantics></math></inline-formula>. The extended adjacency matrix <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mrow><mi>e</mi><mi>x</mi></mrow></msub><mrow><mo>(</mo><mi mathvariant="script">H</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> is an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula> matrix defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mrow><mi>e</mi><mi>x</mi></mrow></msub><mrow><mo>(</mo><mi mathvariant="script">H</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><msub><mi>b</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>b</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mfenced separators="" open="(" close=")"><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>d</mi><mi>i</mi></msub><msub><mi>d</mi><mi>j</mi></msub></mfrac></mstyle><mo>+</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>d</mi><mi>j</mi></msub><msub><mi>d</mi><mi>i</mi></msub></mfrac></mstyle></mfenced></mrow></semantics></math></inline-formula>, whenever <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>i</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>v</mi><mi>j</mi></msub></semantics></math></inline-formula> are adjacent and equal to zero otherwise. The largest eigenvalue of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mrow><mi>e</mi><mi>x</mi></mrow></msub><mrow><mo>(</mo><mi mathvariant="script">H</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is called the extended adjacency spectral radius of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula> and the sum of the absolute values of its eigenvalues is called the extended adjacency energy of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula>. In this paper, we obtain some sharp upper and lower bounds for the extended adjacency spectral radius in terms of different graph parameters and characterize the extremal graphs attaining these bounds. We also obtain some new bounds for the extended adjacency energy of a graph and characterize the extremal graphs attaining these bounds. In both cases, we show our bounds are better than some already known bounds in the literature.https://www.mdpi.com/2078-2489/15/10/586graphseigenvaluesspectral radiusextended adjacency eigenvaluesextended adjacency energy |
| spellingShingle | Alaa Altassan Hilal A. Ganie Yilun Shang On the Extended Adjacency Eigenvalues of a Graph Information graphs eigenvalues spectral radius extended adjacency eigenvalues extended adjacency energy |
| title | On the Extended Adjacency Eigenvalues of a Graph |
| title_full | On the Extended Adjacency Eigenvalues of a Graph |
| title_fullStr | On the Extended Adjacency Eigenvalues of a Graph |
| title_full_unstemmed | On the Extended Adjacency Eigenvalues of a Graph |
| title_short | On the Extended Adjacency Eigenvalues of a Graph |
| title_sort | on the extended adjacency eigenvalues of a graph |
| topic | graphs eigenvalues spectral radius extended adjacency eigenvalues extended adjacency energy |
| url | https://www.mdpi.com/2078-2489/15/10/586 |
| work_keys_str_mv | AT alaaaltassan ontheextendedadjacencyeigenvaluesofagraph AT hilalaganie ontheextendedadjacencyeigenvaluesofagraph AT yilunshang ontheextendedadjacencyeigenvaluesofagraph |