We construct quantum fields and vertex algebras on star graphs. Our construction uses a specific deformation of the canonical algebra, encoding the interaction at the vertex of the graph. Special attention is devoted to the scale invariant interactions, which determine the critical properties of the system. We classify the critical points and investigate their features. The physical observables, we focus on, are the Casimir energy density and the conductance. Among the examples illustrating nontrivial interactions in the bulk of the graph, we consider the nonlinear Schrodinger and the massless Thirring models.