On the <i>A<sub>α</sub></i>-Eigenvalues of Signed Graphs
For <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>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/16/1990 |
| _version_ | 1797522968514920448 |
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| author | Germain Pastén Oscar Rojo Luis Medina |
| author_facet | Germain Pastén Oscar Rojo Luis Medina |
| author_sort | Germain Pastén |
| collection | DOAJ |
| description | For <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>, let <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 stretchy="false">(</mo><msub><mi>G</mi><mi>σ</mi></msub><mo stretchy="false">)</mo></mrow><mo>=</mo><mi>α</mi><mi>D</mi><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><mo>+</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mi>A</mi><mrow><mo stretchy="false">(</mo><msub><mi>G</mi><mi>σ</mi></msub><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, where <i>G</i> is a simple undirected graph, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> is the diagonal matrix of its vertex degrees and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><msub><mi>G</mi><mi>σ</mi></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> is the adjacency matrix of the signed graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mi>σ</mi></msub></semantics></math></inline-formula> whose underlying graph is <i>G</i>. In this paper, basic properties of <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 stretchy="false">(</mo><msub><mi>G</mi><mi>σ</mi></msub><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> are obtained, its positive semidefiniteness is studied and some bounds on its eigenvalues are derived—in particular, lower and upper bounds on its largest eigenvalue are obtained. |
| first_indexed | 2024-03-10T08:36:49Z |
| format | Article |
| id | doaj.art-22053dfdc9c94361a8108e05bec5eeae |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T08:36:49Z |
| publishDate | 2021-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-22053dfdc9c94361a8108e05bec5eeae2023-11-22T08:34:55ZengMDPI AGMathematics2227-73902021-08-01916199010.3390/math9161990On the <i>A<sub>α</sub></i>-Eigenvalues of Signed GraphsGermain Pastén0Oscar Rojo1Luis Medina2Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta 1240000, ChileDepartamento de Matemáticas, Universidad Católica del Norte, Antofagasta 1240000, ChileDepartamento de Matemáticas, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta 1240000, ChileFor <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>, let <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 stretchy="false">(</mo><msub><mi>G</mi><mi>σ</mi></msub><mo stretchy="false">)</mo></mrow><mo>=</mo><mi>α</mi><mi>D</mi><mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><mo>+</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mi>A</mi><mrow><mo stretchy="false">(</mo><msub><mi>G</mi><mi>σ</mi></msub><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, where <i>G</i> is a simple undirected graph, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> is the diagonal matrix of its vertex degrees and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><msub><mi>G</mi><mi>σ</mi></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> is the adjacency matrix of the signed graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mi>σ</mi></msub></semantics></math></inline-formula> whose underlying graph is <i>G</i>. In this paper, basic properties of <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 stretchy="false">(</mo><msub><mi>G</mi><mi>σ</mi></msub><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> are obtained, its positive semidefiniteness is studied and some bounds on its eigenvalues are derived—in particular, lower and upper bounds on its largest eigenvalue are obtained.https://www.mdpi.com/2227-7390/9/16/1990signed graphadjacency matrixbalanced signed graphswitching equivalent graphseigenvalue bounds |
| spellingShingle | Germain Pastén Oscar Rojo Luis Medina On the <i>A<sub>α</sub></i>-Eigenvalues of Signed Graphs Mathematics signed graph adjacency matrix balanced signed graph switching equivalent graphs eigenvalue bounds |
| title | On the <i>A<sub>α</sub></i>-Eigenvalues of Signed Graphs |
| title_full | On the <i>A<sub>α</sub></i>-Eigenvalues of Signed Graphs |
| title_fullStr | On the <i>A<sub>α</sub></i>-Eigenvalues of Signed Graphs |
| title_full_unstemmed | On the <i>A<sub>α</sub></i>-Eigenvalues of Signed Graphs |
| title_short | On the <i>A<sub>α</sub></i>-Eigenvalues of Signed Graphs |
| title_sort | on the i a sub α sub i eigenvalues of signed graphs |
| topic | signed graph adjacency matrix balanced signed graph switching equivalent graphs eigenvalue bounds |
| url | https://www.mdpi.com/2227-7390/9/16/1990 |
| work_keys_str_mv | AT germainpasten ontheiasubasubieigenvaluesofsignedgraphs AT oscarrojo ontheiasubasubieigenvaluesofsignedgraphs AT luismedina ontheiasubasubieigenvaluesofsignedgraphs |