Some Extremal Graphs with Respect to Sombor Index
Let <i>G</i> be a graph with set of vertices <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow>...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/11/1202 |
| _version_ | 1797532672774373376 |
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| author | Kinkar Chandra Das Yilun Shang |
| author_facet | Kinkar Chandra Das Yilun Shang |
| author_sort | Kinkar Chandra Das |
| collection | DOAJ |
| description | Let <i>G</i> be a graph with set of vertices <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>=</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula> and edge set <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>. Very recently, a new degree-based molecular structure descriptor, called Sombor index is denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> and is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>=</mo><mi>S</mi><mi>O</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msub><mi>v</mi><mi>i</mi></msub><msub><mi>v</mi><mi>j</mi></msub><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></munder></mstyle><mspace width="0.166667em"></mspace><msqrt><mrow><msub><mi>d</mi><mi>G</mi></msub><msup><mrow><mo>(</mo><msub><mi>v</mi><mi>i</mi></msub><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msub><mi>d</mi><mi>G</mi></msub><msup><mrow><mo>(</mo><msub><mi>v</mi><mi>j</mi></msub><mo>)</mo></mrow><mn>2</mn></msup></mrow></msqrt></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi>G</mi></msub><mrow><mo>(</mo><msub><mi>v</mi><mi>i</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is 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> in <i>G</i>. In this paper we present some lower and upper bounds on the Sombor index of graph <i>G</i> in terms of graph parameters (clique number, chromatic number, number of pendant vertices, etc.) and characterize the extremal graphs. |
| first_indexed | 2024-03-10T11:03:38Z |
| format | Article |
| id | doaj.art-947feb469bd9483ea7293d67ef90eeb6 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T11:03:38Z |
| publishDate | 2021-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-947feb469bd9483ea7293d67ef90eeb62023-11-21T21:22:02ZengMDPI AGMathematics2227-73902021-05-01911120210.3390/math9111202Some Extremal Graphs with Respect to Sombor IndexKinkar Chandra Das0Yilun Shang1Department of Mathematics, Sungkyunkwan University, Suwon 16419, KoreaDepartment of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UKLet <i>G</i> be a graph with set of vertices <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>=</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula> and edge set <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>. Very recently, a new degree-based molecular structure descriptor, called Sombor index is denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> and is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>=</mo><mi>S</mi><mi>O</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msub><mi>v</mi><mi>i</mi></msub><msub><mi>v</mi><mi>j</mi></msub><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></munder></mstyle><mspace width="0.166667em"></mspace><msqrt><mrow><msub><mi>d</mi><mi>G</mi></msub><msup><mrow><mo>(</mo><msub><mi>v</mi><mi>i</mi></msub><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msub><mi>d</mi><mi>G</mi></msub><msup><mrow><mo>(</mo><msub><mi>v</mi><mi>j</mi></msub><mo>)</mo></mrow><mn>2</mn></msup></mrow></msqrt></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi>G</mi></msub><mrow><mo>(</mo><msub><mi>v</mi><mi>i</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is 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> in <i>G</i>. In this paper we present some lower and upper bounds on the Sombor index of graph <i>G</i> in terms of graph parameters (clique number, chromatic number, number of pendant vertices, etc.) and characterize the extremal graphs.https://www.mdpi.com/2227-7390/9/11/1202graphSombor indexchromatic numberclique number |
| spellingShingle | Kinkar Chandra Das Yilun Shang Some Extremal Graphs with Respect to Sombor Index Mathematics graph Sombor index chromatic number clique number |
| title | Some Extremal Graphs with Respect to Sombor Index |
| title_full | Some Extremal Graphs with Respect to Sombor Index |
| title_fullStr | Some Extremal Graphs with Respect to Sombor Index |
| title_full_unstemmed | Some Extremal Graphs with Respect to Sombor Index |
| title_short | Some Extremal Graphs with Respect to Sombor Index |
| title_sort | some extremal graphs with respect to sombor index |
| topic | graph Sombor index chromatic number clique number |
| url | https://www.mdpi.com/2227-7390/9/11/1202 |
| work_keys_str_mv | AT kinkarchandradas someextremalgraphswithrespecttosomborindex AT yilunshang someextremalgraphswithrespecttosomborindex |