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Materials made of graphyne, graphyne oxide, and graphyne quantum dots have drawn a lot of interest due to their potential uses in medicinal nanotechnology. Their remarkable physical, chemical, and mechanical qualities, which make them very desirable for a variety of prospective purposes in this area, are mostly to blame for this. In the subject of mathematical chemistry, molecular topology deals with the algebraic characterization of molecules. Molecular descriptors can examine a compound's properties and describe its molecular topology. By evaluating these indices, researchers can predict a molecule's behavior including its reactivity, solubility, and toxicity. Amidst the captivating realm of carbon allotropes, γ -graphyne has emerged as a mesmerizing tool, with exquisite attention due to its extraordinary electronic, optical, and mechanical attributes. Research into its possible applications across numerous scientific and technological fields has increased due to this motivated attention. The exploration of molecular descriptors for characterizing γ -graphyne is very attractive. As a result, it is crucial to investigate and predict γ -graphyne's molecular topology in order to comprehend its physicochemical characteristics fully. In this regard, various characterizations of γ -graphyne and zigzag γ -graphyne nanoribbons, by computing and comparing distance-degree-based topological indices, leap Zagreb indices, hyper leap Zagreb indices, leap gourava indices, and hyper leap gourava indices, are investigated. [ABSTRACT FROM AUTHOR]