Abstract:

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.

Key words: residual graph, minimum order, extremal graph