A Unified Approach for Extremal General Exponential Multiplicative Zagreb Indices
The study of the maximum and minimal characteristics of graphs is the focus of the significant field of mathematics known as extreme graph theory. Finding the biggest or smallest graphs that meet specified criteria is the main goal of this discipline. There are several applications of extremal graph...
| Main Authors: | , , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-07-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/12/7/675 |
| _version_ | 1797590312861827072 |
|---|---|
| author | Rashad Ismail Muhammad Azeem Yilun Shang Muhammad Imran Ali Ahmad |
| author_facet | Rashad Ismail Muhammad Azeem Yilun Shang Muhammad Imran Ali Ahmad |
| author_sort | Rashad Ismail |
| collection | DOAJ |
| description | The study of the maximum and minimal characteristics of graphs is the focus of the significant field of mathematics known as extreme graph theory. Finding the biggest or smallest graphs that meet specified criteria is the main goal of this discipline. There are several applications of extremal graph theory in various fields, including computer science, physics, and chemistry. Some of the important applications include: Computer networking, social networking, chemistry and physics as well. Recently, in 2021 exponential multiplicative Zagreb indices were introduced. In generalization, we introduce the generalized form of exponential multiplicative Zagreb indices for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mo>+</mo></msup><mo>\</mo><mrow><mo>{</mo><mn>1</mn><mo>}</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> Furthermore, to see the behaviour of generalized first and second exponential Zagreb indices for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mo>+</mo></msup><mo>\</mo><mrow><mo>{</mo><mn>1</mn><mo>}</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> we used a transformation method. In term of the two newly developed generalized exponential multiplicative Zagreb indices, we will investigate the extremal bicyclic, uni-cyclic and trees graphs. Four graph transformations are used and some bounds are presented in terms of generalized exponential multiplicative Zagreb indices. |
| first_indexed | 2024-03-11T01:18:49Z |
| format | Article |
| id | doaj.art-a16321fd5d894184a5571f4ee40343b6 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T01:18:49Z |
| publishDate | 2023-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-a16321fd5d894184a5571f4ee40343b62023-11-18T18:17:53ZengMDPI AGAxioms2075-16802023-07-0112767510.3390/axioms12070675A Unified Approach for Extremal General Exponential Multiplicative Zagreb IndicesRashad Ismail0Muhammad Azeem1Yilun Shang2Muhammad Imran3Ali Ahmad4Department of Mathematics, Faculty of Science and Arts, Mahayl Assir, King Khalid University, Abha 61421, Saudi ArabiaDepartment of Mathematics, Riphah International University, Lahore 54000, PakistanDepartment of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UKDepartment of Mathematics, Riphah International University, Lahore 54000, PakistanDepartment of Information Technology and Security, College of Computer Science and Information Technology, Jazan University, Jazan 45142, Saudi ArabiaThe study of the maximum and minimal characteristics of graphs is the focus of the significant field of mathematics known as extreme graph theory. Finding the biggest or smallest graphs that meet specified criteria is the main goal of this discipline. There are several applications of extremal graph theory in various fields, including computer science, physics, and chemistry. Some of the important applications include: Computer networking, social networking, chemistry and physics as well. Recently, in 2021 exponential multiplicative Zagreb indices were introduced. In generalization, we introduce the generalized form of exponential multiplicative Zagreb indices for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mo>+</mo></msup><mo>\</mo><mrow><mo>{</mo><mn>1</mn><mo>}</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> Furthermore, to see the behaviour of generalized first and second exponential Zagreb indices for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mo>+</mo></msup><mo>\</mo><mrow><mo>{</mo><mn>1</mn><mo>}</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> we used a transformation method. In term of the two newly developed generalized exponential multiplicative Zagreb indices, we will investigate the extremal bicyclic, uni-cyclic and trees graphs. Four graph transformations are used and some bounds are presented in terms of generalized exponential multiplicative Zagreb indices.https://www.mdpi.com/2075-1680/12/7/675extremal graphsfirst and second generalized exponential multiplicative Zagreb indicesunified approachgraph transformations |
| spellingShingle | Rashad Ismail Muhammad Azeem Yilun Shang Muhammad Imran Ali Ahmad A Unified Approach for Extremal General Exponential Multiplicative Zagreb Indices Axioms extremal graphs first and second generalized exponential multiplicative Zagreb indices unified approach graph transformations |
| title | A Unified Approach for Extremal General Exponential Multiplicative Zagreb Indices |
| title_full | A Unified Approach for Extremal General Exponential Multiplicative Zagreb Indices |
| title_fullStr | A Unified Approach for Extremal General Exponential Multiplicative Zagreb Indices |
| title_full_unstemmed | A Unified Approach for Extremal General Exponential Multiplicative Zagreb Indices |
| title_short | A Unified Approach for Extremal General Exponential Multiplicative Zagreb Indices |
| title_sort | unified approach for extremal general exponential multiplicative zagreb indices |
| topic | extremal graphs first and second generalized exponential multiplicative Zagreb indices unified approach graph transformations |
| url | https://www.mdpi.com/2075-1680/12/7/675 |
| work_keys_str_mv | AT rashadismail aunifiedapproachforextremalgeneralexponentialmultiplicativezagrebindices AT muhammadazeem aunifiedapproachforextremalgeneralexponentialmultiplicativezagrebindices AT yilunshang aunifiedapproachforextremalgeneralexponentialmultiplicativezagrebindices AT muhammadimran aunifiedapproachforextremalgeneralexponentialmultiplicativezagrebindices AT aliahmad aunifiedapproachforextremalgeneralexponentialmultiplicativezagrebindices AT rashadismail unifiedapproachforextremalgeneralexponentialmultiplicativezagrebindices AT muhammadazeem unifiedapproachforextremalgeneralexponentialmultiplicativezagrebindices AT yilunshang unifiedapproachforextremalgeneralexponentialmultiplicativezagrebindices AT muhammadimran unifiedapproachforextremalgeneralexponentialmultiplicativezagrebindices AT aliahmad unifiedapproachforextremalgeneralexponentialmultiplicativezagrebindices |