  In the present paper, we apply Alexandrov geometry methods to study geometric
analysis aspects of infinite semiplanar graphs with nonnegative combinatorial
curvature in the sense of Higuchi. We obtain the metric classification of these
graphs and construct the graphs embedded in the projective plane minus one
point. Moreover, we show the volume doubling property and the Poincar\'e
inequality on such graphs. The quadratic volume growth of these graphs implies
the parabolicity. In addition, we prove the polynomial growth harmonic function
theorem analogous to the case of Riemannian manifolds.
