  Quasisymmetry is a well-studied property of homeomorphisms between metric
spaces, and Ahlfors regular conformal dimension is a quasisymmetric invariant.
In the present paper, we consider the Ahlfors regular conformal dimension of
metrics on infinite graphs, and show that this notion coincides with the
critical exponent of $p$-energies. Moreover, we give a relation between the
Ahlfors regular conformal dimension and the spectral dimension of a graph.
