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We present a cup product formula on the Hochschild cohomology of a family of quiver algebras. We use the formula to determine the set of homogeneous non-nilpotent Hochschild cocycles and construct a canonical isomorphism between Hochschild cohomology modulo nilpotents and a subalgebra ofthat is not finitely generated. For some members of the family, we present the Gerstenhaber ideal of homogeneous nilpotent cocycles using homotopy lifting technique. We then determine their Hochschild cohomology modulo the weak Gerstenhaber ideal generated by nilpotent elements, thereby providing an answer to a question of Reiner Hermann.

Cup product and Gerstenhaber bracket on Hochschild cohomology of a family of quiver algebras