Doubly stochastic matrix has many important applications, the family of compact graphs can be seen as the generalization of the famous Birkhoff theorem which is about doubly stochastic matrix,  and is of important research value． Determine whether a graph is a compact graph is a difficult problem,  at present there are only few compact graphs known． This paper gives two important results: the graph constructed by any compact graph combining some isolated points is a
compact graph; the graph constructed by adding one pendant edge to each vertex of any compact graph is also a compact graph． By these two results,  we can construct an infinite number of compact graph family from already known compact graph．