The least doubling constant of a path graph

TL;DR

This paper investigates the minimal doubling constant for measures on path graphs, determining exact values for infinite and finite cases, and analyzing the structure of optimal measures.

Contribution

It provides explicit formulas for the least doubling constant on path graphs and characterizes the structure of measures that minimize this constant.

Findings

For G=Z, the least doubling constant is 3.

For finite path graphs L_n, the constant is between 1+2cos(pi/(n+1)) and 3.

Optimal measures are characterized for L_n and Z.

Abstract

We study the least doubling constant $C_G$ among all possible doubling measures defined on a path graph $G$. We consider both finite and infinite cases and show that, if $G=\mathbb Z$, $C_{\mathbb Z}=3$, while for $G=L_n$, the path graph with $n$ vertices, one has $1+2\cos(\frac{\pi}{n+1})\leq C_{L_n}<3$, with equality on the lower bound if and only if $n\le8$. Moreover, we analyze the structure of doubling minimizers on $L_n$ and $\mathbb Z$, those measures whose doubling constant is the smallest possible.