  On a finite connected metric graph, we establish upper bounds for the
eigenvalues of the Laplacian. These bounds depend on the length, the Betti
number, and the number of pendant vertices. For trees, these estimates are
sharp. We also establish sharp upper bounds for the spectral gap of the
complete graph $K_4$. The proofs are based on estimates for eigenvalues on
graphs with Dirichlet conditions imposed at the pendant vertices.
