# Seminario

#### Seminario Análisis y Aplicaciones

Least doubling constants on graphs and spectral theory

We study the least doubling constant $$C_G$$ among all possible doubling measures defined on a graph $$G$$. In particular, for a path graph $$G=L_n$$,  we show that  $$1+2\cos(\frac{\pi}{n+1})\leq C_{L_n}<3$$, with equality on the lower bound if and only if $$n\le8$$.
We then prove  that $$C_G$$, for a general graph $$G$$, can be estimated from below by  $$1+ r(A_G)$$, where $$r(A_G)$$ is the spectral radius of the adjacency matrix of  $$G$$, and study when both quantities coincide.