On the <i>A</i><sub>α</sub>-Spectral Radii of Cactus Graphs
Let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>A</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> be the adjacent matrix and <inline-f...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/6/869 |
| _version_ | 1827716020173799424 |
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| author | Chunxiang Wang Shaohui Wang Jia-Bao Liu Bing Wei |
| author_facet | Chunxiang Wang Shaohui Wang Jia-Bao Liu Bing Wei |
| author_sort | Chunxiang Wang |
| collection | DOAJ |
| description | Let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>A</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> be the adjacent matrix and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>D</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> the diagonal matrix of the degrees of a graph <i>G</i>, respectively. For <inline-formula> <math display="inline"> <semantics> <mrow> <mn>0</mn> <mo>≤</mo> <mi>α</mi> <mo>≤</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, the <inline-formula> <math display="inline"> <semantics> <msub> <mi>A</mi> <mi>α</mi> </msub> </semantics> </math> </inline-formula>-matrix is the general adjacency and signless Laplacian spectral matrix having the form of <inline-formula> <math 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>. Clearly, <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>A</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is the adjacent matrix and <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2</mn> <msub> <mi>A</mi> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msub> </mrow> </semantics> </math> </inline-formula> is the signless Laplacian matrix. A cactus is a connected graph such that any two of its cycles have at most one common vertex, that is an extension of the tree. The <inline-formula> <math display="inline"> <semantics> <msub> <mi>A</mi> <mi>α</mi> </msub> </semantics> </math> </inline-formula>-spectral radius of a cactus graph with <i>n</i> vertices and <i>k</i> cycles is explored. The outcomes obtained in this paper can imply some previous bounds from trees to cacti. In addition, the corresponding extremal graphs are determined. Furthermore, we proposed all eigenvalues of such extremal cacti. Our results extended and enriched previous known results. |
| first_indexed | 2024-03-10T19:31:44Z |
| format | Article |
| id | doaj.art-a90bb74d260146b68ced11b25451a943 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T19:31:44Z |
| publishDate | 2020-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-a90bb74d260146b68ced11b25451a9432023-11-20T02:06:13ZengMDPI AGMathematics2227-73902020-05-018686910.3390/math8060869On the <i>A</i><sub>α</sub>-Spectral Radii of Cactus GraphsChunxiang Wang0Shaohui Wang1Jia-Bao Liu2Bing Wei3School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, ChinaDepartment of Mathematics, Louisiana College, Pineville, LA 71359, USASchool of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, ChinaDepartment of Mathematics, University of Mississippi, University, MS 38677, USALet <inline-formula> <math display="inline"> <semantics> <mrow> <mi>A</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> be the adjacent matrix and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>D</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> the diagonal matrix of the degrees of a graph <i>G</i>, respectively. For <inline-formula> <math display="inline"> <semantics> <mrow> <mn>0</mn> <mo>≤</mo> <mi>α</mi> <mo>≤</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, the <inline-formula> <math display="inline"> <semantics> <msub> <mi>A</mi> <mi>α</mi> </msub> </semantics> </math> </inline-formula>-matrix is the general adjacency and signless Laplacian spectral matrix having the form of <inline-formula> <math 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>. Clearly, <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>A</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is the adjacent matrix and <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2</mn> <msub> <mi>A</mi> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msub> </mrow> </semantics> </math> </inline-formula> is the signless Laplacian matrix. A cactus is a connected graph such that any two of its cycles have at most one common vertex, that is an extension of the tree. The <inline-formula> <math display="inline"> <semantics> <msub> <mi>A</mi> <mi>α</mi> </msub> </semantics> </math> </inline-formula>-spectral radius of a cactus graph with <i>n</i> vertices and <i>k</i> cycles is explored. The outcomes obtained in this paper can imply some previous bounds from trees to cacti. In addition, the corresponding extremal graphs are determined. Furthermore, we proposed all eigenvalues of such extremal cacti. Our results extended and enriched previous known results.https://www.mdpi.com/2227-7390/8/6/869signless Laplacianadjacency matrixtreecacti |
| spellingShingle | Chunxiang Wang Shaohui Wang Jia-Bao Liu Bing Wei On the <i>A</i><sub>α</sub>-Spectral Radii of Cactus Graphs Mathematics signless Laplacian adjacency matrix tree cacti |
| title | On the <i>A</i><sub>α</sub>-Spectral Radii of Cactus Graphs |
| title_full | On the <i>A</i><sub>α</sub>-Spectral Radii of Cactus Graphs |
| title_fullStr | On the <i>A</i><sub>α</sub>-Spectral Radii of Cactus Graphs |
| title_full_unstemmed | On the <i>A</i><sub>α</sub>-Spectral Radii of Cactus Graphs |
| title_short | On the <i>A</i><sub>α</sub>-Spectral Radii of Cactus Graphs |
| title_sort | on the i a i sub α sub spectral radii of cactus graphs |
| topic | signless Laplacian adjacency matrix tree cacti |
| url | https://www.mdpi.com/2227-7390/8/6/869 |
| work_keys_str_mv | AT chunxiangwang ontheiaisubasubspectralradiiofcactusgraphs AT shaohuiwang ontheiaisubasubspectralradiiofcactusgraphs AT jiabaoliu ontheiaisubasubspectralradiiofcactusgraphs AT bingwei ontheiaisubasubspectralradiiofcactusgraphs |