Summary: The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. The edge grafting operation on a graph is a kind of edge moving between two vertices of the graph. In this paper, we introduce two new edge grafting operations and show how the graph energy changes under these edge grafting operations. Let $G(n)$ be the set of all unicyclic graphs with n vertices. Using these edge grafting operations and the Coulson integral formula for the energy of a monic real polynomial, we characterize the unicyclic graphs with the first to the seventh minimal energies in $G(n)$ ($n \geq 11$).