Two extensions of Leech labeling to the class of all graphs

Abstract

Let [Formula: see text] be a tree of order n and let [Formula: see text] 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 [Formula: see text] If the set of weights of the [Formula: see text] paths in T is [Formula: see text] 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 present two extensions of Leech labeling to the class of all graphs, namely, Leech labeling and geodesic Leech labeling. For this purpose we introduce two parameters [Formula: see text] and [Formula: see text] where [Formula: see text] is the total number of paths in G and [Formula: see text] is the total number of geodesic paths in G. We determine the values of these parameters for some graph classes. We present several families of graphs which are Leech graphs, geodesic Leech graphs and also graphs which are not Leech graphs. A list of open problems is also included.

Keywords

• Leech labeling
• Leech graph
• geodesic Leech graph
• 05C78