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Topological index is an important numerical magnitude that can reflect the whole structure of a graph. Degree-based indices are mathematical descriptors worked in chemical graph theory to compute the connectivity and underlying descriptions of molecules. These indices, such as the Wiener index, Randi´ c index, and Zagreb indices, are obtained from the quantities of vertices in molecular graphs. The Wiener index correspond to the calculation of gaps between all pairs of nodes in a graph, presenting information about molecular dimension and separating. Zagreb indices, including the first and second Zagreb indices, summarize the measure allocation and node connectivity within a molecular graph. Degree-based indices portray a critical position in molecular modeling, QSAR investigations, and medication layout, requiring perceptions into molecular regional anatomy and forecasting various physicochemical assets. In this article, we compute topological descriptors for complete p-ary trees. Interesting comparison of these indices are is also shown in tabular and graphical format. Moreover, expressions for multiple Zagreb indices and polynomials for these important classes are found.

Metal-Organic Frameworks have already shown enormous possibilities, starting from catalysis to drug delivery. This paper outlines the topological indices in the case of tetra-cyano-benzene transition metal organic networks due to the unique properties that they exhibit in electronic, magnetic, and catalytic applications. A detailed mathematical investigation of topological indices such as atom-bond connectivity, geometric arithmetic index, and others is presented in relation to tetra-cyano-benzene and its complexation with transition metals.