### Two classes of non-Leech trees

#### Abstract

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 ^{n}C_{2} paths in *T* are exactly 1, 2,...,^{n}C_{2}, 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 *P _{n}* =(

*v*

_{1},

*v*

_{2},...,

*v*) by attaching a pendent vertex at

_{n}*v*

_{n-1}is not a Leech tree for all

*n*≥ 4.

#### Keywords

#### Full Text:

PDFDOI: http://dx.doi.org/10.5614/ejgta.2020.8.1.15

#### References

B. Calhoun, K. Ferland, L. Lister and J. Polhill, Minimal distinct distance trees, J. Combin. Math. Combin. Comput. 61 (2007), 33–57.

G. Chartrand and L.Lesniak, Graphs and digraphs, CRC (2005).

D. Leach, Modular Leech trees of order atmost 8, J. Combin., (2014), Article ID 218086.

D. Leach and M. Walsh, Generalized Leech trees, J. Combin. Math. Combin. Comput. 78 (2011), 15–22.

J. Leech, Another tree labeling problem, Amer. Math. Monthly 82 (1975), 923–925.

M. Ozen, H. Wang, D. Yalman, Note on Leech-type questions of tree, Integers 16 (2016),#A21

L.A. Szekely, H. Wang and Y. Zhang, Some non-existence results on Leech trees, Bull. Inst. Combin. Appl. 44 (2005), 37–45.

H. Taylor, Odd path sums in an edge-labeled tree, Math. Magazine 50 (5) (1977), 258–259.

H. Taylor, A distinct distance set of 9 nodes in a tree of diameter 36, Discrete Math. 93 (1991), 167–168.

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