The spectral dimension of random brushes

We consider a class of random graphs, called random brushes, which are constructed by adding linear graphs of random lengths to the vertices of Zd viewed as a graph. We prove that for d = 2 all random brushes have spectral dimension ds = 2. For d = 3, we have 5/2 <= ds <= 3, and for d ≥ 4 we have 3 ≤ ds ≤ d