A Balancing Theorem for Spanning Trees of Rectangular Grid Graphs

Abstract: We prove that, among rectangular grid graphs with a fixed number of vertices, the number of spanning trees increases when the side lengths are made more balanced. In particular, among all rectangular grid graphs with $n^2$ vertices, the square $n\times n$ grid has the largest number of spanning trees. The proof starts with the Laplacian product formula, passes to hyperbolic coordinates, and compares logarithms by separating a discrete-concavity term from a positive decreasing residual term.