A finiteness theorem for geodesic Leech wheels

Abstract

Let f be a labeling of the edges of a finite graph G by positive integers, and let the weight of a path be the sum of the labels of its edges. The labeling is a geodesic Leech labeling if the weights of the geodesics are exactly 1, 2, ..., t_gp(G), each occurring once, where t_gp(G) is the geodesic path number of G. Let W_n be the wheel on n vertices, a hub joined to an (n-1)-cycle. Our main result is an upper bound: if n >= 5 and W_n is geodesic Leech, then n <= 40. The proof quantifies, via a finite Fourier kernel, the Sidon-type structure of the spoke labels, in which only the cyclically adjacent pairs are allowed as defects, and closes the last three cases with a six-variable Parseval argument. In the other direction, explicit labelings of W_7, ..., W_13, found by a computer search, answer in the negative a problem of Lakshmanan S. and Manattu, who had found labelings of W_5 and W_6 and expected every W_n with n >= 7 to be a non-geodesic Leech graph. Writing E for the set of n >= 5 for which W_n is geodesic Leech, we obtain {5, 6, ..., 13} is contained in E, which is contained in {5, 6, ..., 40}.