Center coloring Cc(G) is a kind of coloring that is to color the vertices of a graph G is such a way that if vertices have different distances from the center then they must receive different colors. Two adjacent vertices can receive the same color. In this paper we observed the behavior of the center coloring under several graph operations such as adding, removing, union, join, sequential join, Cartesian product, corona product of graphs and we get some bounds for center coloring number in these graph operations.