  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 $\dim_\mathrm{ARC}\le d_s<2.$
