Leech index of a tree

• Published in: Journal of discrete mathematical sciences & cryptography Vol. 25; no. 8; pp. 2237 - 2247
• Main Authors: Varghese, Seena, Lakshmanan, Aparna, Arumugam, S.
• Format: Journal Article
• Language: English
• Published: Taylor & Francis 17.11.2022
• Subjects: Leech index; Leech labeling; Leech tree
• ISSN: 0972-0529; 2169-0065
• DOI: 10.1080/09720529.2020.1800217

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.