A Turán-type extremal problem for the number of spanning trees in $C_4$-free graphs

Abstract: For a graph (F), the Turán number (\ex(n,F)) is the maximum number of edges in an (F)-free graph on (n) vertices. Let (q\ge 2) be an integer and set (n=q^{2}+q+1). Brown and Erdős, Rényi and Sós independently proved that $\ex(n,C_{4})\ge \frac12 q(q+1)<sup>{2}$ for every prime power (q), and Füredi subsequently established the upper bound $\frac12 q(q+1)<sup>{2}$ for $\ex(n,C_{4})$ whenever (q\notin{1, 7,9,11,13}). In this article, we prove that every (C_{4})-free graph (G) on (n) vertices with at most (\frac12 q(q+1)^{2}) edges satisfies $τ(G)\le n<sup>{(n-3)/2}$, where (τ(G)) denotes the number of spanning trees of (G). In particular, for every prime power $q\notin{7,9,11,13}$, the above upper bound on $τ(G)$ is attained precisely by the orthogonal polarity graphs, thereby proving London's conjecture for all such $q$.