Ad-Hoc Lanzhou Index
This paper initiates the study of the mathematical aspects of the ad-hoc Lanzhou index. If <i>G</i> is a graph with the vertex set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{<...
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MDPI AG
2023-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/20/4256 |
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| author | Akbar Ali Yilun Shang Darko Dimitrov Tamás Réti |
| author_facet | Akbar Ali Yilun Shang Darko Dimitrov Tamás Réti |
| author_sort | Akbar Ali |
| collection | DOAJ |
| description | This paper initiates the study of the mathematical aspects of the ad-hoc Lanzhou index. If <i>G</i> is a graph with the vertex set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>x</mi><mi>n</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula>, then the ad-hoc Lanzhou index of <i>G</i> is defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mrow><mi>L</mi><mi>z</mi></mrow><mo>˜</mo></mover><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>d</mi><mi>i</mi></msub><msup><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>−</mo><msub><mi>d</mi><mi>i</mi></msub><mo>)</mo></mrow><mn>2</mn></msup></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>d</mi><mi>i</mi></msub></semantics></math></inline-formula> represents the degree of the vertex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>x</mi><mi>i</mi></msub></semantics></math></inline-formula>. Several identities for the ad-hoc Lanzhou index, involving some existing topological indices, are established. The problems of finding graphs with the extremum values of the ad-hoc Lanzhou index from the following sets of graphs are also attacked: (i) the set of all connected <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula>-cyclic graphs of a fixed order, (ii) the set of all connected molecular <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula>-cyclic graphs of a fixed order, (iii) the set of all graphs of a fixed order, and (iv) the set of all connected molecular graphs of a fixed order. |
| first_indexed | 2024-03-10T21:04:51Z |
| format | Article |
| id | doaj.art-97a788d7c96a49419852bb91d28cc661 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T21:04:51Z |
| publishDate | 2023-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-97a788d7c96a49419852bb91d28cc6612023-11-19T17:13:24ZengMDPI AGMathematics2227-73902023-10-011120425610.3390/math11204256Ad-Hoc Lanzhou IndexAkbar Ali0Yilun Shang1Darko Dimitrov2Tamás Réti3Department of Mathematics, College of Science, University of Ha’il, Ha’il P.O. Box 2240, Saudi ArabiaDepartment of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UKFaculty of Information Studies, 8000 Novo Mesto, SloveniaDonát Bánki Faculty of Mechanical and Safety Engineering, Óbuda University, H-1034 Budapest, HungaryThis paper initiates the study of the mathematical aspects of the ad-hoc Lanzhou index. If <i>G</i> is a graph with the vertex set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>x</mi><mi>n</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula>, then the ad-hoc Lanzhou index of <i>G</i> is defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mrow><mi>L</mi><mi>z</mi></mrow><mo>˜</mo></mover><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>d</mi><mi>i</mi></msub><msup><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>−</mo><msub><mi>d</mi><mi>i</mi></msub><mo>)</mo></mrow><mn>2</mn></msup></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>d</mi><mi>i</mi></msub></semantics></math></inline-formula> represents the degree of the vertex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>x</mi><mi>i</mi></msub></semantics></math></inline-formula>. Several identities for the ad-hoc Lanzhou index, involving some existing topological indices, are established. The problems of finding graphs with the extremum values of the ad-hoc Lanzhou index from the following sets of graphs are also attacked: (i) the set of all connected <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula>-cyclic graphs of a fixed order, (ii) the set of all connected molecular <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula>-cyclic graphs of a fixed order, (iii) the set of all graphs of a fixed order, and (iv) the set of all connected molecular graphs of a fixed order.https://www.mdpi.com/2227-7390/11/20/4256topological indexchemical graph theoryad-hoc Lanzhou indexLanzhou indexforgotten topological coindex |
| spellingShingle | Akbar Ali Yilun Shang Darko Dimitrov Tamás Réti Ad-Hoc Lanzhou Index Mathematics topological index chemical graph theory ad-hoc Lanzhou index Lanzhou index forgotten topological coindex |
| title | Ad-Hoc Lanzhou Index |
| title_full | Ad-Hoc Lanzhou Index |
| title_fullStr | Ad-Hoc Lanzhou Index |
| title_full_unstemmed | Ad-Hoc Lanzhou Index |
| title_short | Ad-Hoc Lanzhou Index |
| title_sort | ad hoc lanzhou index |
| topic | topological index chemical graph theory ad-hoc Lanzhou index Lanzhou index forgotten topological coindex |
| url | https://www.mdpi.com/2227-7390/11/20/4256 |
| work_keys_str_mv | AT akbarali adhoclanzhouindex AT yilunshang adhoclanzhouindex AT darkodimitrov adhoclanzhouindex AT tamasreti adhoclanzhouindex |