Two classes of non-Leech trees - Seena Varghese; Aparna Lakshmanan S; Subramanian Arumugam (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2020)
Let T be a tree of order n. For any edge labeling f : E → 1,2,3,... the weight of a path P is the sum of the labels of the edges of P and is denoted by w(P). If the weights of the nC2 paths in T are exactly 1, 2,...,nC2, then f is called a Leech labeling and a tree which admits a Leech labeling is called a Leech tree. In this paper we determine all Leech trees having diameter three. We also prove that the tree obtained from the path Pn =(v1,v2,...,vn) by attaching a pendent vertex at vn-1 is not a Leech tree for all n ≥ 4.