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We study zero-one laws for random graphs. We focus on the following questionthat was asked by many: Given a graph property P, is there a language of graphsable to express P while obeying the zero-one law? Our results show that on theone hand there is a (regular) language able to express connectivity andk-colorability for any constant k and still obey the zero-one law. On the otherhand we show that in any (semiregular) language strong enough to expressHamiltonicity one can interpret arithmetic and thus the zero-one law failsmiserably. This answers a question of Blass and Harary.

Random graphs and Lindstrom quantifiers for natural graph properties