Extremal Graphs to Vertex Degree Function Index for Convex Functions
The <i>vertex-degree function index</i> <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>f</mi></msub><mrow><mo>(</mo>&l...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-12-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/1/31 |
| _version_ | 1797446159468331008 |
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| author | Dong He Zhen Ji Chenxu Yang Kinkar Chandra Das |
| author_facet | Dong He Zhen Ji Chenxu Yang Kinkar Chandra Das |
| author_sort | Dong He |
| collection | DOAJ |
| description | The <i>vertex-degree function index</i> <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>f</mi></msub><mrow><mo>(</mo><mi mathvariant="sans-serif">Γ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>f</mi></msub><mrow><mo>(</mo><mi mathvariant="sans-serif">Γ</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi mathvariant="sans-serif">Γ</mi><mo>)</mo></mrow></msub><mi>f</mi><mrow><mo>(</mo><mi>d</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for a function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> defined on non-negative real numbers. In this paper, we determine the extremal graphs with the maximum (minimum) vertex degree function index in the set of all <i>n</i>-vertex chemical trees, trees, and connected graphs. We also present the Nordhaus–Gaddum-type results for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>f</mi></msub><mrow><mo>(</mo><mi mathvariant="sans-serif">Γ</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>H</mi><mi>f</mi></msub><mrow><mo>(</mo><mover><mi mathvariant="sans-serif">Γ</mi><mo>¯</mo></mover><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>f</mi></msub><mrow><mo>(</mo><mi mathvariant="sans-serif">Γ</mi><mo>)</mo></mrow><mo>·</mo><msub><mi>H</mi><mi>f</mi></msub><mrow><mo>(</mo><mover><mi mathvariant="sans-serif">Γ</mi><mo>¯</mo></mover><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-09T13:37:20Z |
| format | Article |
| id | doaj.art-9b0ad7e809394f66b18e8d7671f8f01a |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T13:37:20Z |
| publishDate | 2022-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-9b0ad7e809394f66b18e8d7671f8f01a2023-11-30T21:11:21ZengMDPI AGAxioms2075-16802022-12-011213110.3390/axioms12010031Extremal Graphs to Vertex Degree Function Index for Convex FunctionsDong He0Zhen Ji1Chenxu Yang2Kinkar Chandra Das3School of Mathematics and Statistis, Qinghai Normal University, Xining 810008, ChinaSchool of Computer, Qinghai Normal University, Xining 810008, ChinaSchool of Computer, Qinghai Normal University, Xining 810008, ChinaDepartment of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of KoreaThe <i>vertex-degree function index</i> <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>f</mi></msub><mrow><mo>(</mo><mi mathvariant="sans-serif">Γ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>f</mi></msub><mrow><mo>(</mo><mi mathvariant="sans-serif">Γ</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi mathvariant="sans-serif">Γ</mi><mo>)</mo></mrow></msub><mi>f</mi><mrow><mo>(</mo><mi>d</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for a function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> defined on non-negative real numbers. In this paper, we determine the extremal graphs with the maximum (minimum) vertex degree function index in the set of all <i>n</i>-vertex chemical trees, trees, and connected graphs. We also present the Nordhaus–Gaddum-type results for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>f</mi></msub><mrow><mo>(</mo><mi mathvariant="sans-serif">Γ</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>H</mi><mi>f</mi></msub><mrow><mo>(</mo><mover><mi mathvariant="sans-serif">Γ</mi><mo>¯</mo></mover><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>f</mi></msub><mrow><mo>(</mo><mi mathvariant="sans-serif">Γ</mi><mo>)</mo></mrow><mo>·</mo><msub><mi>H</mi><mi>f</mi></msub><mrow><mo>(</mo><mover><mi mathvariant="sans-serif">Γ</mi><mo>¯</mo></mover><mo>)</mo></mrow></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2075-1680/12/1/31vertex degree function indextreecemical treeconnected graphNordhaus–Gaddum-type result |
| spellingShingle | Dong He Zhen Ji Chenxu Yang Kinkar Chandra Das Extremal Graphs to Vertex Degree Function Index for Convex Functions Axioms vertex degree function index tree cemical tree connected graph Nordhaus–Gaddum-type result |
| title | Extremal Graphs to Vertex Degree Function Index for Convex Functions |
| title_full | Extremal Graphs to Vertex Degree Function Index for Convex Functions |
| title_fullStr | Extremal Graphs to Vertex Degree Function Index for Convex Functions |
| title_full_unstemmed | Extremal Graphs to Vertex Degree Function Index for Convex Functions |
| title_short | Extremal Graphs to Vertex Degree Function Index for Convex Functions |
| title_sort | extremal graphs to vertex degree function index for convex functions |
| topic | vertex degree function index tree cemical tree connected graph Nordhaus–Gaddum-type result |
| url | https://www.mdpi.com/2075-1680/12/1/31 |
| work_keys_str_mv | AT donghe extremalgraphstovertexdegreefunctionindexforconvexfunctions AT zhenji extremalgraphstovertexdegreefunctionindexforconvexfunctions AT chenxuyang extremalgraphstovertexdegreefunctionindexforconvexfunctions AT kinkarchandradas extremalgraphstovertexdegreefunctionindexforconvexfunctions |