We study the magnetic properties of nanometer-sized graphene structures with triangular and hexagonal shapes terminated by zig-zag edges. We discuss how the shape of the island, the imbalance in the number of atoms belonging to the two graphene sublattices, the existence of zero-energy states, and the total and local magnetic moment are intimately related. We consider electronic interactions both in a mean-field approximation of the one-orbital Hubbard model and with density functional calculations. Both descriptions yield values for the ground state total spin, $S$, consistent with Liebs theorem for bipartite lattices. Triangles have a finite $S$ for all sizes whereas hexagons have S=0 and develop local moments above a critical size of $approx 1.5$ nm.
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