Minimum Zagreb Eccentricity Indices of Two-Mode Network with Applications in Boiling Point and Benzenoid Hydrocarbons
A two-mode network is a type of network in which nodes can be divided into two sets in such a way that links can be established between different types of nodes. The relationship between two separate sets of entities can be modeled as a bipartite network. In computer networks data is transmitted in...
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MDPI AG
2022-04-01
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| author | Ali Al Khabyah Shahid Zaman Ali N. A. Koam Ali Ahmad Asad Ullah |
| author_facet | Ali Al Khabyah Shahid Zaman Ali N. A. Koam Ali Ahmad Asad Ullah |
| author_sort | Ali Al Khabyah |
| collection | DOAJ |
| description | A two-mode network is a type of network in which nodes can be divided into two sets in such a way that links can be established between different types of nodes. The relationship between two separate sets of entities can be modeled as a bipartite network. In computer networks data is transmitted in form of packets between source to destination. Such packet-switched networks rely on routing protocols to select the best path. Configurations of these protocols depends on the network acquirements; that is why one routing protocol might be efficient for one network and may be inefficient for a other. Because some protocols deal with hop-count (number of nodes in the path) while others deal with distance vector. This paper investigates the minimum transmission in two-mode networks. Based on some parameters, we obtained the minimum transmission between the class of all connected n-nodes in bipartite networks. These parameters are helpful to modify or change the path of a given network. Furthermore, by using least squares fit, we discussed some numerical results of the regression model of the boiling point in benzenoid hydrocarbons. The results show that the correlation of the boiling point in benzenoid hydrocarbons of the first Zagreb eccentricity index gives better result as compare to the correlation of second Zagreb eccentricity index. In case of a connected network, the first Zagreb eccentricity index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ξ</mi><mn>1</mn></msub><mrow><mo>(</mo><mo>ℵ</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is defined as the sum of the square of eccentricities of the nodes, and the second Zagreb eccentricity index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ξ</mi><mn>2</mn></msub><mrow><mo>(</mo><mo>ℵ</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is defined as the sum of the product of eccentricities of the adjacent nodes. This article deals with the minimum transmission with respect to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ξ</mi><mi>i</mi></msub><mrow><mo>(</mo><mo>ℵ</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></semantics></math></inline-formula> among all <i>n</i>-node extremal bipartite networks with given matching number, diameter, node connectivity and link connectivity. |
| first_indexed | 2024-03-10T03:56:58Z |
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| spelling | doaj.art-d13b4e1597aa4aeeb23b0ac0c72eb4762023-11-23T08:43:37ZengMDPI AGMathematics2227-73902022-04-01109139310.3390/math10091393Minimum Zagreb Eccentricity Indices of Two-Mode Network with Applications in Boiling Point and Benzenoid HydrocarbonsAli Al Khabyah0Shahid Zaman1Ali N. A. Koam2Ali Ahmad3Asad Ullah4Department of Mathematics, College of Science, Jazan University, New Campus, Jazan 2097, Saudi ArabiaDepartment of Mathematics, University of Sialkot, Sialkot 51310, PakistanDepartment of Mathematics, College of Science, Jazan University, New Campus, Jazan 2097, Saudi ArabiaCollege of Computer Science and Information Technology, Jazan University, Jazan 2097, Saudi ArabiaDepartment of Mathematical Sciences, Karakoram International University Gilgit, Gilgit 15100, PakistanA two-mode network is a type of network in which nodes can be divided into two sets in such a way that links can be established between different types of nodes. The relationship between two separate sets of entities can be modeled as a bipartite network. In computer networks data is transmitted in form of packets between source to destination. Such packet-switched networks rely on routing protocols to select the best path. Configurations of these protocols depends on the network acquirements; that is why one routing protocol might be efficient for one network and may be inefficient for a other. Because some protocols deal with hop-count (number of nodes in the path) while others deal with distance vector. This paper investigates the minimum transmission in two-mode networks. Based on some parameters, we obtained the minimum transmission between the class of all connected n-nodes in bipartite networks. These parameters are helpful to modify or change the path of a given network. Furthermore, by using least squares fit, we discussed some numerical results of the regression model of the boiling point in benzenoid hydrocarbons. The results show that the correlation of the boiling point in benzenoid hydrocarbons of the first Zagreb eccentricity index gives better result as compare to the correlation of second Zagreb eccentricity index. In case of a connected network, the first Zagreb eccentricity index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ξ</mi><mn>1</mn></msub><mrow><mo>(</mo><mo>ℵ</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is defined as the sum of the square of eccentricities of the nodes, and the second Zagreb eccentricity index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ξ</mi><mn>2</mn></msub><mrow><mo>(</mo><mo>ℵ</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is defined as the sum of the product of eccentricities of the adjacent nodes. This article deals with the minimum transmission with respect to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ξ</mi><mi>i</mi></msub><mrow><mo>(</mo><mo>ℵ</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></semantics></math></inline-formula> among all <i>n</i>-node extremal bipartite networks with given matching number, diameter, node connectivity and link connectivity.https://www.mdpi.com/2227-7390/10/9/1393Zagreb eccentricity indicesbipartite networksmatching numberdiameternode connectivitylink connectivity |
| spellingShingle | Ali Al Khabyah Shahid Zaman Ali N. A. Koam Ali Ahmad Asad Ullah Minimum Zagreb Eccentricity Indices of Two-Mode Network with Applications in Boiling Point and Benzenoid Hydrocarbons Mathematics Zagreb eccentricity indices bipartite networks matching number diameter node connectivity link connectivity |
| title | Minimum Zagreb Eccentricity Indices of Two-Mode Network with Applications in Boiling Point and Benzenoid Hydrocarbons |
| title_full | Minimum Zagreb Eccentricity Indices of Two-Mode Network with Applications in Boiling Point and Benzenoid Hydrocarbons |
| title_fullStr | Minimum Zagreb Eccentricity Indices of Two-Mode Network with Applications in Boiling Point and Benzenoid Hydrocarbons |
| title_full_unstemmed | Minimum Zagreb Eccentricity Indices of Two-Mode Network with Applications in Boiling Point and Benzenoid Hydrocarbons |
| title_short | Minimum Zagreb Eccentricity Indices of Two-Mode Network with Applications in Boiling Point and Benzenoid Hydrocarbons |
| title_sort | minimum zagreb eccentricity indices of two mode network with applications in boiling point and benzenoid hydrocarbons |
| topic | Zagreb eccentricity indices bipartite networks matching number diameter node connectivity link connectivity |
| url | https://www.mdpi.com/2227-7390/10/9/1393 |
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