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Counting Anosov graphs

In recent work by Dani and Mainkar, a family of finite simple graphs was used to construct nilmanifolds admitting Anosov diffeomorphisms. Our main object of study is this particular set of graphs, which we call Anosov graphs. Moreover, Dani and Mainkar give a lower bound on the number of Anosov graphs in terms of the number of vertices and number of edges. In this work, we improve this lower bound in terms of vertices and edges, and we give lower and upper bounds solely in terms of the number of vertices.