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In this paper, we study Young Fibonacci graphs Gn, a special family of graphsthat are constructed with the help of integer partitions. Young diagrams are also used in the construction of graphs. The family of graphs is rich in structure. Thus, we investigate various properties of the family of graphs which include degree based structure and topological in-dices. Topological indices like Zagreb Index, Wiener Index, Randic Index and Connective Eccentricity Index of these graphs are computed. We also study the eigenvalues and energy of the graph