symmetry S S Article Study of θ φ Networks via Zagreb Connection Indices Muhammad Asif 1 , Bartlomiej Kizielewicz 2 , Atiq ur Rehman 1 , Muhammad Hussain 1 and Wojciech Salabun 2, *   Citation: Asif, M.; Kizielewicz, B.; Rehman, A.u.; Hussain, M.; Salabun, W. Study of θ φ Networks via Zagreb Connection Indices. Symmetry 2021, 13, 1991. https://doi.org/10.3390/ sym13111991 Academic Editor: Serge Lawrencenko Received: 19 August 2021 Accepted: 8 October 2021 Published: 21 October 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Department of Mathematics, COMSATS University Islamabad (Lahore Campus), Lahore 53713, Pakistan; sp18-pmt-001@cuilahore.edu.pk (M.A.); atiqurehman@cuilahore.edu.pk (A.u.R.); muhammad.hussain@cuilahore.edu.pk (M.H.) 2 Research Team on Intelligent Decision Support Systems, Department of Artificial Intelligence and Applied Mathematics, Faculty of Computer Science and Information Technology, West Pomeranian University of Technology in Szczecin, ul. ˙ Zolnierska 49, 71-210 Szczecin, Poland; bartlomiej-kizielewicz@zut.edu.pl * Correspondence: wojciech.salabun@zut.edu.pl; Tel.: +48-91-449-5580 Abstract: Graph theory can be used to optimize interconnection network systems. The compatibility of such networks mainly depends on their topology. Topological indices may characterize the topology of such networks. In this work, we studied a symmetric network θ φ formed by φ time repetition of the process of joining θ copies of a selected graph Ω in such a way that corresponding vertices of Ω in all the copies are joined with each other by a new edge. The symmetry of θ φ is ensured by the involvement of complete graph K θ in the construction process. The free hand to choose an initial graph Ω and formation of chemical graphs using θ φ Ω enhance its importance as a family of graphs which covers all the pre-defined graphs, along with space for new graphs, possibly formed in this way. We used Zagreb connection indices for the characterization of θ φ Ω. These indices have gained worth in the field of chemical graph theory in very small duration due to their predictive power for enthalpy, entropy, and acentric factor. These computations are mathematically novel and assist in topological characterization of θ φ Ω to enable its emerging use. Keywords: Zagreb connection indices; graph invariants; interconnection networks; mk graphs; topological index 1. Introduction Graph theory provides a fundamental tool for designing and analyzing desired net- works with accuracy and gives a thorough understanding of the manners by which the parts of a system interconnected through topology of an interconnection network [1]. Along with the other disciplines, graph theory has a special place in the field of chemistry, especially in chemical graph theory [2]. Thus, chemical graph theory is a composition of chemistry, computer science, and graph theory [3–5]. It provides information about organic substances regarding their physicochemical properties with the help of graph invariants using chemical graphs associated with their molecular structure. A chemical graph is a simple connected and hydrogen depleted graph consisting of vertices replacing atoms and edges for the bonds between atoms. A simple graph is comprised of only a single edge between two vertices and no self-loop (an edge with the same initial and final vertex). Graph invariants have strong applications in quantitative structure properties relation- ship (QSPR) investigation [6]. These invariants reduce the practical work to some extent to study the new chemicals structures using the topology of desired chemical structure. Topological indices are also the graph invariants that map chemical graphs into a numeric value and characterize the underlying structure’s topology. Harry Wiener, in 1947, first introduced Wiener index [7]. Later on, the first and second Zagreb indices were proposed in Reference [8,9] as Symmetry 2021, 13, 1991. https://doi.org/10.3390/sym13111991 https://www.mdpi.com/journal/symmetry