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In 2008, J. Skowronek-kazi o ´ w extended the study of the clique number ω G Z n to the zero-divisor graph of the ring Z n , but their result was imperfect. In this paper, we reconsider ω G Z n of the ring Z n and give some counterexamples. We propose a constructive method for calculating ω G Z n and give an algorithm for calculating the clique number of zero-divisor graph. Furthermore, we consider the case of the ternary zero-divisor and give the generation algorithm of the ternary zero-divisor graphs.

In 1988, I.Beck showed that the chromatic number of G(Z n ) is equal to its clique number.In 2004, S.Akbari and A.Mohammadian proved that the edge chromatic number of G(Z n ) is equal to its maximum degree,in 2008, J.Skowronek-kaziow give formulas calculating the clique number and the maximum degree of G(Z n ) ,but he have a error about clique number of G(Z n ) , we consider the zero-divisor graph G(Z n ) of the ring Z n .we give formulas calculating the clique number of G(Z n ).We present a constructed method to calculate the clique number.