Let \(G\) be a finite abelian group of order \(n\). The barycentric Ramsey number \(BR(H,G)\) is the minimum positive integer \(r\) such that any coloring of the edges of the complete graph \(K_r\) by elements of \(G\) contains a subgraph \(H\) whose assigned edge colors constitute a barycentric sequence, i.e., there exists one edge whose color is the “average” of the colors of all edges in \(H\). When the number of edges \(e(H) \equiv 0 \pmod{\exp(G)}\), \(BR(H,G)\) are the well-known zero-sum Ramsey numbers \(R(H,G)\). In this work, these Ramsey numbers are determined for some graphs, in particular, for graphs with five edges without isolated vertices using \(G = \mathbb{Z}_n\), where \(2 \leq n \leq 4\), and for some graphs \(H\) with \(e(H) \equiv 0 \pmod{2}\) using \(G = \mathbb{Z}_2^s\).