Doubling constants and spectral theory on graphs

We study the least doubling constant among all possible measures defined on a (finite or infinite) graph G. show that this can be estimated from below by 1+r(AG), where r(AG) is spectral radius of adjacency matrix G, and when both quantities coincide. also illustrate how amenability automorphism group related to finding minimizers. Finally, we give complete characterization graphs with smaller than 3, in spirit Smith graphs.