Spanning-Tree Extremality in $C_4$-Free Graphs

Abstract

We study the maximum number of spanning trees in connected $n$-vertex $C_4$-free graphs. For projective-plane orders $n=q^2+q+1$, we determine the spanning-tree count of every polarity graph and show that polarity graphs with exactly $q+1$ absolute points maximize this count within the polarity family, attaining $n^{(n-3)/2}$ spanning trees. Combined with a stability theorem of He, Ma and Yang \cite{HeMaYangCSIAM23}, this yields the same exact upper bound for all sufficiently dense $C_4$-free graphs in the known polarity stability regime. For arbitrary $C_4$-free graphs at these orders, we derive a global upper bound implying \[ \log \mathrm{st}(n,C_4)=\frac{n-3}{2}\log n+O(\sqrt n), \] where $\mathrm{st}(n,C_4)$ denotes the maximum number of spanning trees over connected $n$-vertex $C_4$-free graphs.

7 pages. An updated version of the manuscript submitted previously