The least doubling constant of a path graph

Abstract

We study the least doubling constant CG among all possible doubling measures defined on a path graph G. We consider both finite and infinite cases and show that, if G = Z, CZ = 3, while for G = Ln, the path graph with n vertices, one has 1 + 2 cos( π / n+1 ) ≤ CLn < 3, with equality on the lower bound if and only if n ≤ 8. Moreover, we analyze the structure of doubling minimizers on Ln and Z, those measures whose doubling constant is the smallest possible.