Some New Bounds for <i>α</i>-Adjacency Energy of Graphs
Let <i>G</i> be a graph with the adjacency matrix <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></mr...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/9/2173 |
| _version_ | 1797602194146459648 |
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| author | Haixia Zhang Zhuolin Zhang |
| author_facet | Haixia Zhang Zhuolin Zhang |
| author_sort | Haixia Zhang |
| collection | DOAJ |
| description | Let <i>G</i> be a graph with the adjacency matrix <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 let <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> be the diagonal matrix of the degrees of <i>G</i>. Nikiforov first defined the matrix <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></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></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>α</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>, which shed new light on <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>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>, and yielded some surprises. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>−</mo></mrow></semantics></math></inline-formula>adjacency energy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>E</mi><msub><mi>A</mi><mi>α</mi></msub></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of <i>G</i> is a new invariant that is calculated from the eigenvalues 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>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In this work, by combining matrix theory and the graph structure properties, we provide some upper and lower bounds for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>E</mi><msub><mi>A</mi><mi>α</mi></msub></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> in terms of graph parameters (the order <i>n</i>, the edge size <i>m</i>, etc.) and characterize the corresponding extremal graphs. In addition, we obtain some relations between <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>E</mi><msub><mi>A</mi><mi>α</mi></msub></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and other energies such as the energy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Some results can be applied to appropriately estimate the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-adjacency energy using some given graph parameters rather than by performing some tedious calculations. |
| first_indexed | 2024-03-11T04:13:36Z |
| format | Article |
| id | doaj.art-d6265d06fc2c4cac94ae510f56a7cb53 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T04:13:36Z |
| publishDate | 2023-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-d6265d06fc2c4cac94ae510f56a7cb532023-11-17T23:20:57ZengMDPI AGMathematics2227-73902023-05-01119217310.3390/math11092173Some New Bounds for <i>α</i>-Adjacency Energy of GraphsHaixia Zhang0Zhuolin Zhang1Department of Mathematics, Taiyuan University of Science and Technology, Taiyuan 030024, ChinaDepartment of Mathematics, Taiyuan University of Science and Technology, Taiyuan 030024, ChinaLet <i>G</i> be a graph with the adjacency matrix <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 let <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> be the diagonal matrix of the degrees of <i>G</i>. Nikiforov first defined the matrix <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></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></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>α</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>, which shed new light on <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>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>, and yielded some surprises. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>−</mo></mrow></semantics></math></inline-formula>adjacency energy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>E</mi><msub><mi>A</mi><mi>α</mi></msub></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of <i>G</i> is a new invariant that is calculated from the eigenvalues 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>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In this work, by combining matrix theory and the graph structure properties, we provide some upper and lower bounds for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>E</mi><msub><mi>A</mi><mi>α</mi></msub></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> in terms of graph parameters (the order <i>n</i>, the edge size <i>m</i>, etc.) and characterize the corresponding extremal graphs. In addition, we obtain some relations between <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>E</mi><msub><mi>A</mi><mi>α</mi></msub></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and other energies such as the energy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Some results can be applied to appropriately estimate the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-adjacency energy using some given graph parameters rather than by performing some tedious calculations.https://www.mdpi.com/2227-7390/11/9/2173adjacency matrixenergyα-adjacency matrixα-adjacency energy |
| spellingShingle | Haixia Zhang Zhuolin Zhang Some New Bounds for <i>α</i>-Adjacency Energy of Graphs Mathematics adjacency matrix energy α-adjacency matrix α-adjacency energy |
| title | Some New Bounds for <i>α</i>-Adjacency Energy of Graphs |
| title_full | Some New Bounds for <i>α</i>-Adjacency Energy of Graphs |
| title_fullStr | Some New Bounds for <i>α</i>-Adjacency Energy of Graphs |
| title_full_unstemmed | Some New Bounds for <i>α</i>-Adjacency Energy of Graphs |
| title_short | Some New Bounds for <i>α</i>-Adjacency Energy of Graphs |
| title_sort | some new bounds for i α i adjacency energy of graphs |
| topic | adjacency matrix energy α-adjacency matrix α-adjacency energy |
| url | https://www.mdpi.com/2227-7390/11/9/2173 |
| work_keys_str_mv | AT haixiazhang somenewboundsforiaiadjacencyenergyofgraphs AT zhuolinzhang somenewboundsforiaiadjacencyenergyofgraphs |