m-K_{n}-残差图是由P. Erd\"{o}s, F. Harary和M. Klawe等人提出的, 当m=1时, 他们证明了当n\neq1,2,3,4时, K_{n+1}\timesK_{2}是唯一的具有最小阶的连通的K_{n}- 残差图. 首先得到了m-K_{n}-残差图的重要性质, 同时证明了当n=1,2,3,4时, 连通K_{n}-残差图的最小阶和极图, 其中当n=1,2时得到唯一极图; 当n=3,4时, 证明了恰有两个不同构的极图, 从而彻底解决连通的K_{n}-残差图的最小阶和极图问题. 最后证明了当n\neq1,2,3,4时, K_{n+1}\timesK_{2}是唯一的具有最小阶的连通的K_{n}-残差图.

The definition of m-K_{n}-residual graph was raised by P. Erd\"{o}s, F. Harary and M. Klawe. When n\neq 1,2,3,4, they proved that K_{n+1}\times K_{2} is only connected to K_{n}-residual graph which has minimum order. In this paper, we have studied m-K_{n}-residual graph, and obtained some important properties. At the same time, we proved that  the connected K_{n}-residual graph of the minimum order and the extremal graph for n=1,2,3,4. When n=1,2, it is the only extremal graph. When n=3,4, we proved just two connected residual graph  non isomorphic with the minimum order, so as to thoroughly solve the connected K_{n}-residual graph of the minimum order and extremal graph's problems. Finally we prove that K_{n+1}\times K_{2} is only connected with the minimum order of K_{n}-residual graph, when n\neq 1,2,3,4.