The spectral dimension $d_s$ of a weighted graph is an exponent associated with the asymptotic behavior of the random walk on the graph. The Ahlfors regular conformal dimension $\dim_\mathrm{ ARC}$ of the graph distance is a quasisymmetric invariant, where quasisymmetry is a well-studied property of homeomorphisms between metric spaces. In this paper, we give a typical example of a fractal-like graph with $d_s<\dim_\mathrm{ ARC}<2$ and prove a sufficient condition for $\dim_\mathrm{ ARC}\le d_s<2.$
Categories: math.PR