On the Maximum Number of Spanning Trees in $C_4$-Free Graphs

We introduce a ``Kirchhoff--Tur\'an'' variant of the extremal $C_4$ problem: among all simple connected $n$-vertex $C_4$-free graphs $G$, maximize the number of spanning trees $\tau(G)$. For the projective-plane orders $n=q^2+q+1$ we compute an exact formula for the Erd\H{o}s--R\'enyi orthogonal polarity graph $ER_q$, namely $\tau(ER_q)=n^{(n-3)/2}$, via a polarity spectral identity and Kirchhoff's matrix--tree theorem. We also give an explicit general upper bound on $\mathrm{st}(n,C_4)$ at these $n$ using a sharp degree-sequence inequality for $\tau(G)$ and a degree-balancing argument; this matches the lower bound in the leading exponential term.