Counting Anosov graphs

TL;DR

This paper investigates a special class of graphs called Anosov graphs, improving bounds on their quantity based on vertices and edges, which has implications for constructing certain geometric structures.

Contribution

It provides improved lower bounds on the number of Anosov graphs based on vertices and edges, and establishes bounds solely in terms of vertices.

Findings

Improved lower bounds on the number of Anosov graphs.

Bounds established solely in terms of vertices.

Enhanced understanding of the combinatorial structure of Anosov graphs.

Abstract

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.