We revise our former definition of graph operations and correspondingly adapt the construction of graph term algebras. As a first contribution to a prospective research field, Universal Graph Algebra, we generalize some basic concepts and results from algebras to graph algebras. To tackle this generalization task, we revise and reformulate traditional set-theoretic definitions, constructions and proofs in Universal Algebra by means of more category-theoretic concepts and constructions. In particular, we generalize the concept of generated subalgebra and prove that all monomorphic homomorphisms between graph algebras are regular. Derived graph operations are the other main topic. After an in-depth analysis of terms as representations of derived operations in traditional algebras, we identify three basic mechanisms to construct new graph operations out of given ones: parallel composition, instantiation, and sequential composition. As a counterpart of terms, we introduce graph operation expressions with a structure as close as possible to the structure of terms. We show that the three mechanisms allow us to construct, for any graph operation expression, a corresponding derived graph operation in any graph algebra.