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 Z^d viewed as a graph. We prove that for d=2 all random brushes have spectral dimension d_s=2. For d=3 we have {5\over 2}\leq d_s\leq 3 and for d\geq 4 we have 3\leq d_s\leq d.