On General Reduced Second Zagreb Index of Graphs
Graph-based molecular structure descriptors (often called “topological indices”) are useful for modeling the physical and chemical properties of molecules, designing pharmacologically active compounds, detecting environmentally hazardous substances, etc. The graph invariant <inline-formula><...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/19/3553 |
| _version_ | 1827654003290275840 |
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| author | Lkhagva Buyantogtokh Batmend Horoldagva Kinkar Chandra Das |
| author_facet | Lkhagva Buyantogtokh Batmend Horoldagva Kinkar Chandra Das |
| author_sort | Lkhagva Buyantogtokh |
| collection | DOAJ |
| description | Graph-based molecular structure descriptors (often called “topological indices”) are useful for modeling the physical and chemical properties of molecules, designing pharmacologically active compounds, detecting environmentally hazardous substances, etc. The graph invariant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>R</mi><msub><mi>M</mi><mi>α</mi></msub></mrow></semantics></math></inline-formula>, known under the name general reduced second Zagreb index, is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>R</mi><msub><mi>M</mi><mi>α</mi></msub><mrow><mo>(</mo><mi mathvariant="normal">Γ</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi mathvariant="normal">Γ</mi><mo>)</mo></mrow></msub><mrow><mo>(</mo><msub><mi>d</mi><mi mathvariant="normal">Γ</mi></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>+</mo><mi>α</mi><mo>)</mo></mrow><mrow><mo>(</mo><msub><mi>d</mi><mi mathvariant="normal">Γ</mi></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>+</mo><mi>α</mi><mo>)</mo></mrow></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 mathvariant="normal">Γ</mi></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the degree of the vertex <i>v</i> of the graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Γ</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> is any real number. In this paper, among all trees of order <i>n</i>, and all unicyclic graphs of order <i>n</i> with girth <i>g</i>, we characterize the extremal graphs with respect to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>R</mi><msub><mi>M</mi><mi>α</mi></msub></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>α</mi><mo>≥</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo></mrow></semantics></math></inline-formula>. Using the extremal unicyclic graphs, we obtain a lower bound on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>R</mi><msub><mi>M</mi><mi>α</mi></msub><mrow><mo>(</mo><mi mathvariant="normal">Γ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of graphs in terms of order <i>n</i> with <i>k</i> cut edges, and completely determine the corresponding extremal graphs. Moreover, we obtain several upper bounds on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>R</mi><msub><mi>M</mi><mi>α</mi></msub></mrow></semantics></math></inline-formula> of different classes of graphs in terms of order <i>n</i>, size <i>m</i>, independence number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>, chromatic number <i>k</i>, etc. In particular, we present an upper bound on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>R</mi><msub><mi>M</mi><mi>α</mi></msub></mrow></semantics></math></inline-formula> of connected triangle-free graph of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>></mo><mn>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> edges with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>></mo><mo>−</mo><mn>1.5</mn></mrow></semantics></math></inline-formula>, and characterize the extremal graphs. Finally, we prove that the Turán graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> gives the maximum <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>R</mi><msub><mi>M</mi><mi>α</mi></msub><mspace width="0.166667em"></mspace><mrow><mo>(</mo><mi>α</mi><mo>≥</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> among all graphs of order <i>n</i> with chromatic number <i>k</i>. |
| first_indexed | 2024-03-09T21:28:32Z |
| format | Article |
| id | doaj.art-6584896dddc24f7ca25adaa9b4efafd9 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T21:28:32Z |
| publishDate | 2022-09-01 |
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| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-6584896dddc24f7ca25adaa9b4efafd92023-11-23T21:03:30ZengMDPI AGMathematics2227-73902022-09-011019355310.3390/math10193553On General Reduced Second Zagreb Index of GraphsLkhagva Buyantogtokh0Batmend Horoldagva1Kinkar Chandra Das2Department of Mathematics, Mongolian National University of Education, Baga Toiruu-14, Ulaanbaatar 210648, MongoliaDepartment of Mathematics, Mongolian National University of Education, Baga Toiruu-14, Ulaanbaatar 210648, MongoliaDepartment of Mathematics, Sungkyunkwan University, Suwon 16419, KoreaGraph-based molecular structure descriptors (often called “topological indices”) are useful for modeling the physical and chemical properties of molecules, designing pharmacologically active compounds, detecting environmentally hazardous substances, etc. The graph invariant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>R</mi><msub><mi>M</mi><mi>α</mi></msub></mrow></semantics></math></inline-formula>, known under the name general reduced second Zagreb index, is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>R</mi><msub><mi>M</mi><mi>α</mi></msub><mrow><mo>(</mo><mi mathvariant="normal">Γ</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi mathvariant="normal">Γ</mi><mo>)</mo></mrow></msub><mrow><mo>(</mo><msub><mi>d</mi><mi mathvariant="normal">Γ</mi></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>+</mo><mi>α</mi><mo>)</mo></mrow><mrow><mo>(</mo><msub><mi>d</mi><mi mathvariant="normal">Γ</mi></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>+</mo><mi>α</mi><mo>)</mo></mrow></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 mathvariant="normal">Γ</mi></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the degree of the vertex <i>v</i> of the graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Γ</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> is any real number. In this paper, among all trees of order <i>n</i>, and all unicyclic graphs of order <i>n</i> with girth <i>g</i>, we characterize the extremal graphs with respect to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>R</mi><msub><mi>M</mi><mi>α</mi></msub></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>α</mi><mo>≥</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo></mrow></semantics></math></inline-formula>. Using the extremal unicyclic graphs, we obtain a lower bound on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>R</mi><msub><mi>M</mi><mi>α</mi></msub><mrow><mo>(</mo><mi mathvariant="normal">Γ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of graphs in terms of order <i>n</i> with <i>k</i> cut edges, and completely determine the corresponding extremal graphs. Moreover, we obtain several upper bounds on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>R</mi><msub><mi>M</mi><mi>α</mi></msub></mrow></semantics></math></inline-formula> of different classes of graphs in terms of order <i>n</i>, size <i>m</i>, independence number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>, chromatic number <i>k</i>, etc. In particular, we present an upper bound on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>R</mi><msub><mi>M</mi><mi>α</mi></msub></mrow></semantics></math></inline-formula> of connected triangle-free graph of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>></mo><mn>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> edges with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>></mo><mo>−</mo><mn>1.5</mn></mrow></semantics></math></inline-formula>, and characterize the extremal graphs. Finally, we prove that the Turán graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> gives the maximum <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>R</mi><msub><mi>M</mi><mi>α</mi></msub><mspace width="0.166667em"></mspace><mrow><mo>(</mo><mi>α</mi><mo>≥</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> among all graphs of order <i>n</i> with chromatic number <i>k</i>.https://www.mdpi.com/2227-7390/10/19/3553Zagreb indicesgirthclique numberchromatic numberTurán graph |
| spellingShingle | Lkhagva Buyantogtokh Batmend Horoldagva Kinkar Chandra Das On General Reduced Second Zagreb Index of Graphs Mathematics Zagreb indices girth clique number chromatic number Turán graph |
| title | On General Reduced Second Zagreb Index of Graphs |
| title_full | On General Reduced Second Zagreb Index of Graphs |
| title_fullStr | On General Reduced Second Zagreb Index of Graphs |
| title_full_unstemmed | On General Reduced Second Zagreb Index of Graphs |
| title_short | On General Reduced Second Zagreb Index of Graphs |
| title_sort | on general reduced second zagreb index of graphs |
| topic | Zagreb indices girth clique number chromatic number Turán graph |
| url | https://www.mdpi.com/2227-7390/10/19/3553 |
| work_keys_str_mv | AT lkhagvabuyantogtokh ongeneralreducedsecondzagrebindexofgraphs AT batmendhoroldagva ongeneralreducedsecondzagrebindexofgraphs AT kinkarchandradas ongeneralreducedsecondzagrebindexofgraphs |