Small Mahler Measures From Digraphs

We attach Mahler measures to digraphs and find combinatorial realizations of nearly all of the known low-degree (<= 180) small (< 1.3) one-variable Mahler measures. We find one new such measure not on either of the lists maintained by Mossinghoff and Sac-Epee. Considering limits of sequences of measures attached to families of digraphs, we get combinatorial explanations for 57 of the 61 known irreducible two-variable measures below 1.37.