A New Proof of the Erdős–Simonovits Conjecture on Walks

Let G^n be a graph on n vertices and let w_k(G^n) denote the number of walks of length k in G^n divided by n . Erdős and Simonovits conjectured that w_k(G^n)^t ≥ w_t(G^n)^k when k≥ t and both t and k are odd. In 2018, Sağlam proved this conjecture. We give a new shorter proof of this result.