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

Ahlfors Regular Conformal Dimension of Metrics on Infinite Graphs and Spectral Dimension of the Associated Random Walks