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　certain　kind　of　edge　moving　between　two　pendant　paths　starting　from　the　same　vertex.　In　this　paper　we　show　how　the　graph　energy　changes　under　the　edge　grafting　operations　on　unicyclic　and　bipartite　graphs.　We　also　give　some　applications　of　this　result　on　the　comparison　of　graph　energies　between　unicyclic　or　bipartite　graphs.　(C)　2010　Elsevier　Inc.　All　rights　reserved.