Topological indices play paramount role in defining chemical and structural properties of different compounds. Chemical graph theory is useful for predicting the bioactivity and physicochemical properties through numerical invariants. A broad range of topological indices are studied and used in theoretical chemistry. The eccentricity of a graph is maximum distance from one vertex � to another vertex �. The connectivity based topological indices depict vital role in mathematical chemistry. In this paper, we discuss Benzenoid Hourglass Network and calculate its total eccentricity, average eccentricity, atom-bond connectivity index, eccentricity based Zagreb index and geometric-arithmetic index. Some new helpful closed formulas are developed.