In this paper we get a double-star tree of special labelling distribution by applying the edge-moving theorem(lemma 1) to the star tree.Then we obtain a graceful labeling and prove the gracefulness of any tree with a diameter 5 by moving the edges cleverly.
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陈相兵.直径为5的树的优美性[J].华东交通大学学报,2009,(1). CHEN Xiang-bing. The Gracefulness of the Trees with a Diameter 5[J]. JOURNAL OF EAST CHINA JIAOTONG UNIVERSTTY,2009,(1):