On Geodesic Leech Labeling of Some Graph Classes

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.