Triangle-free graphs with the maximum number of cycles

TL;DR

This paper proves that for sufficiently large graphs, the complete bipartite graph maximizes the number of cycles among all triangle-free graphs with the same number of vertices.

Contribution

It establishes the uniqueness of the complete bipartite graph as the maximum-cycle triangle-free graph for large n.

Findings

Complete bipartite graphs have the maximum number of cycles among triangle-free graphs for n ≥ 141.

The complete bipartite graph is uniquely optimal in this class for large n.

Abstract

It is shown that for $n\geq 141$, among all triangle-free graphs on $n$ vertices, the complete equibipartite graph is the unique triangle-free graph with the greatest number of cycles.