Leech index of a tree

Abstract

Let T = (V, E) be a tree of order n. Let f : E → {1, 2, 3, … } be an injective edge labeling of T. 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 set of weights of the paths in T is , then f is called a Leech labeling of T and a tree which admits a Leech labeling is called a Leech tree. In this paper, we introduce a new parameter called Leech index which gives a measure of how close a tree is towards being a Leech tree. Let f : E → {1, 2, 3, … } be an edge labeling of T such that both f and w are injective. Let S denote the set of all weights of the paths in T. Let k_f be the positive integer such that {1, 2, 3, … , k _f } ⊆ S and k_f + 1 ∉ S. Then k(T) = max k_f, where the maximum is taken over all such edge labelings f is called the Leech index of T. In this paper, we determine the Leech index of several families of trees and obtain bounds for this parameter.

Subject Classification: (2010):

• 05C78

Keywords:

• Leech labeling
• Leech tree
• Leech index