Barycentric subspace analysis (BSA) is introduced for a set of unlabeled graphs, which are graphs with no correspondence between nodes. Identifying each graph by the set of its eigenvalues, the graph spectrum space is defined as a novel and computationally efficient quotient manifold of isospectral graphs. In such a manifold, the notion of BSA is extended. It showcases how BSA can be used as a powerful dimensionality reduction technique for complex data. BSA searches for a subspace of a lower dimension, minimizing the projection of data points on such subspace. As the subspace is identified by a set of reference points, the interpretation is easier than with other dimensionality reduction techniques. BSA is performed and compared with clustering and PCA on a simulated dataset and a real-world dataset of airline company networks.