A Note on the Estrada Index of the <i>A</i><sub>α</sub>-Matrix
Let <i>G</i> be a graph on <i>n</i> vertices. The Estrada index of <i>G</i> is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. V. Nikiforov studied hybrids of <inline-formula><math xmlns="http://www.w3.org/199...
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| Format: | Article |
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MDPI AG
2021-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/8/811 |
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| author | Jonnathan Rodríguez Hans Nina |
| author_facet | Jonnathan Rodríguez Hans Nina |
| author_sort | Jonnathan Rodríguez |
| collection | DOAJ |
| description | Let <i>G</i> be a graph on <i>n</i> vertices. The Estrada index of <i>G</i> is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. V. Nikiforov studied hybrids of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> and defined the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mi>α</mi></msub></semantics></math></inline-formula>-matrix for every real <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> as: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>α</mi><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> In this paper, using a different demonstration technique, we present a way to compare the Estrada index of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mi>α</mi></msub></semantics></math></inline-formula>-matrix with the Estrada index of the adjacency matrix of the graph <i>G</i>. Furthermore, lower bounds for the Estrada index are established. |
| first_indexed | 2024-03-10T12:30:38Z |
| format | Article |
| id | doaj.art-2be7c46aee6d42b99f097dadc3827891 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T12:30:38Z |
| publishDate | 2021-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-2be7c46aee6d42b99f097dadc38278912023-11-21T14:41:08ZengMDPI AGMathematics2227-73902021-04-019881110.3390/math9080811A Note on the Estrada Index of the <i>A</i><sub>α</sub>-MatrixJonnathan Rodríguez0Hans Nina1Departamento de Matemáticas, Facultad de Ciencias Básicas, Universidad de Antofagasta, Av. Angamos 601, Antofagasta 1240000, ChileDepartamento de Matemáticas, Facultad de Ciencias Básicas, Universidad de Antofagasta, Av. Angamos 601, Antofagasta 1240000, ChileLet <i>G</i> be a graph on <i>n</i> vertices. The Estrada index of <i>G</i> is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. V. Nikiforov studied hybrids of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> and defined the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mi>α</mi></msub></semantics></math></inline-formula>-matrix for every real <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> as: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mi>α</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>α</mi><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> In this paper, using a different demonstration technique, we present a way to compare the Estrada index of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mi>α</mi></msub></semantics></math></inline-formula>-matrix with the Estrada index of the adjacency matrix of the graph <i>G</i>. Furthermore, lower bounds for the Estrada index are established.https://www.mdpi.com/2227-7390/9/8/811Estrada indexα-adjacency matrixadjacency matrixLaplacian matrix |
| spellingShingle | Jonnathan Rodríguez Hans Nina A Note on the Estrada Index of the <i>A</i><sub>α</sub>-Matrix Mathematics Estrada index α-adjacency matrix adjacency matrix Laplacian matrix |
| title | A Note on the Estrada Index of the <i>A</i><sub>α</sub>-Matrix |
| title_full | A Note on the Estrada Index of the <i>A</i><sub>α</sub>-Matrix |
| title_fullStr | A Note on the Estrada Index of the <i>A</i><sub>α</sub>-Matrix |
| title_full_unstemmed | A Note on the Estrada Index of the <i>A</i><sub>α</sub>-Matrix |
| title_short | A Note on the Estrada Index of the <i>A</i><sub>α</sub>-Matrix |
| title_sort | note on the estrada index of the i a i sub α sub matrix |
| topic | Estrada index α-adjacency matrix adjacency matrix Laplacian matrix |
| url | https://www.mdpi.com/2227-7390/9/8/811 |
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