On Geodesic Leech Labeling of Some Graph Classes

TL;DR

This paper investigates the properties of geodesic Leech labelings in certain graph classes, establishing which graphs admit such labelings and providing explicit labelings for specific cases.

Contribution

It characterizes which graphs are geodesic Leech graphs, proves that cycles of length at least five are not, and provides explicit labelings for wheel graphs W_5 and W_6.

Findings

Cycles C_n for n ≥ 5 are non-geodesic Leech graphs.

At most three regular complete bipartite graphs are geodesic Leech.

Degree sequence does not characterize geodesic Leech graphs.

Abstract

Let $f:E\rightarrow \{1,2,3,\dots\}$ be an edge labeling of $G$. The geodesic path number of $G$, $t_{gp}(G)$, is the number of geodesic paths in $G$. An edge labeling $f$ is called a geodesic Leech labeling, if the set of weights of the geodesic paths in $G$ is $\{1,2,3,\dots,t_{gp}(G)\}$, where the weight of a path $P$ is the sum of the labels assigned to the edges of $P$. A graph which admits a geodesic Leech labeling is called a geodesic Leech graph. Otherwise, we call it a non-geodesic Leech graph. In this paper, we prove that cycles $C_n$, $n \geq 5$ are non-geodesic Leech graphs. We also prove that there are at most three regular complete bipartite graphs that are geodesic Leech. We show that degree sequence cannot characterize geodesic Leech graphs. The geodesic path number of the wheel graph $W_n$ is obtained and the geodesic Leech labeling of $W_5$ and $W_6$ is given.