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 θϕ 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 θϕ is ensured by the involvement of complete graph Kθ in the construction process. The free hand to choose an initial graph Ω and formation of chemical graphs using θϕΩ 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 θϕΩ. 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 θϕΩ to enable its emerging use.
Highlights
Graph theory provides a fundamental tool for designing and analyzing desired networks 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]
Chemical graph theory is a composition of chemistry, computer science, and graph theory [3,4,5]
Summary
Graph theory provides a fundamental tool for designing and analyzing desired networks 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]. Graph invariants have strong applications in quantitative structure properties relationship (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. Zagreb connection indices studied in Reference [10,11,12,13] are defined as ZC1(Ω) = ∑ τs, s∈V(Ω). The published work of Reference [15,16,17,18], along with the chemical applicability of these indices and formation of chemical networks using θφΩ, provides motivation for the study of θφΩ via Zagreb connection indices. We first compute exact results for Zagreb connection indices ZC1, ZC2, and ZC1∗ of θφΩ for arbitrary values of θ and φ when Ω = N1 consists single vertex. We computed exact results for θφΩ when Ω belongs to a certain family of graphs as applications of computed results
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