  Gamma conjecture I and the underlying Conjecture $\mathcal{O}$ for Fano
manifolds were proposed by Galkin, Golyshev and Iritani recently. We show that
both conjectures hold for all two-dimensional Fano manifolds. We prove
Conjecture $\mathcal{O}$ by deriving a generalized Perron-Frobenius theorem on
eigenvalues of real matrices and a vanishing result of certain Gromov-Witten
invariants for del Pezzo surfaces. We prove Gamma conjecture I by applying
mirror techniques proposed by Galkin-Iritani together with the study of Gamma
conjecture I for weighted projective spaces.
