Zagreb indices of generalized operations on graphs
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Date
2021-03-12
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UMT Lahore
Abstract
Let Γ be a graph with vertex set V (Γ) and edge set E(Γ) ⊆ V (Γ) ×V (Γ). A topological index (TI) is a function from A to the set of real numbers (ℜ) that associates each element of A to a unique real number, where A is a collection of finite, simple and undirected graphs. Harry Wiener (1947) defined the first distance-based TI to determine the boiling point of paraffin. In 1972, Gutman and Trinajesti ́c computed total π-electrons energy of a molecular graph by first degree-based TI called by first Zagreb index. Later on, many distance, degree, and connection number based TIs are derived to predict the physicochemical and structural properties of the graphs such as stability, solubility, surface tension, radius of gyrations, critical temperature and density. In this thesis, for k ∈ N (set of counting numbers), four generalized subdivision-related operations (Sk, Rk, Qk, and Tk) of graphs are defined. Then, using these operations and the concept of the Cartesian product of graphs, the generalized F-sum graphs (Γ1+Fk Γ2) are also defined, where Fk ∈ { Sk, Rk, Qk, and Tk } and Γi are connected graphs for i ∈ {1, 2}. Mainly, various Zagreb indices such as the first Zagreb index, forgotten index, first general Zagreb index, hyper-Zagreb index, second Zagreb index, multiplicative Zagreb indices of the generalized F-sum graphs are computed in the form of their factor graphs. In addition, two different metal-organic networks are defined in their general form by increasing the layers of both the organic ligands and metal nodes. The degree and connection based generalized Zagreb indices are obtained of these networks. Moreover, different Zagreb indices are also computed for these metal-organic networks using their generalized Zagreb indices.