Minimum Zagreb Eccentricity Indices of Two-Mode Network with Applications in Boiling Point and Benzenoid Hydrocarbons

Ali Al Khabyah, Shahid Zaman, Ali N. A. Koam, Ali Ahmad and Asad Ullah
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Ali Al Khabyah: Department of Mathematics, College of Science, Jazan University, New Campus, Jazan 2097, Saudi Arabia
Shahid Zaman: Department of Mathematics, University of Sialkot, Sialkot 51310, Pakistan
Ali N. A. Koam: Department of Mathematics, College of Science, Jazan University, New Campus, Jazan 2097, Saudi Arabia
Ali Ahmad: College of Computer Science and Information Technology, Jazan University, Jazan 2097, Saudi Arabia
Asad Ullah: Department of Mathematical Sciences, Karakoram International University Gilgit, Gilgit 15100, Pakistan

Mathematics, 2022, vol. 10, issue 9, 1-18

Abstract: 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 ξ 1 ( ℵ ) is defined as the sum of the square of eccentricities of the nodes, and the second Zagreb eccentricity index ξ 2 ( ℵ ) is defined as the sum of the product of eccentricities of the adjacent nodes. This article deals with the minimum transmission with respect to ξ i ( ℵ ) , for i = 1 , 2 among all n -node extremal bipartite networks with given matching number, diameter, node connectivity and link connectivity.

Keywords: Zagreb eccentricity indices; bipartite networks; matching number; diameter; node connectivity; link connectivity (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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