Namely, even when an integral representation is known, it turns out that for numerical estimation, our infinite series can be much better than the integral; see, e.g., \[b\].FelkerLyons/ for examples comparing the two approaches.

In order to state our second main result, we recall the Matrix-Tree Theorem. Given a graph $\gh$, define the following matrices indexed by the vertices of $\gh$: Let $D_\gh$ be the diagonal matrix whose $(x, x)$-entry equals the degree of $x$; this is the [**degree matrix**]{}. Let $A_\gh$ be the matrix whose $(x, y)$-entry equals the number of edges joining $x$ and $y$; this is the [**adjacency matrix**]{}. Finally, let $\Delta_\gh := D_\gh - A_\gh$, the [**graph Laplacian matrix**]{}. The Matrix-Tree Theorem says that given any finite graph $\gh$, every cofactor of $\Delta_\gh$ is the same, namely, $\cp(\gh)$ (see, e.g., \[b\].GodRoy/, Lemma 13.2.3). Another version of the Matrix-Tree Theorem states that when $\gh$ is connected, $$\cp(\gh) = |\verts(\gh)|^{-1}
\detp \Delta_\gh
\,,
\label e.MTT$$ where $\detp A$ denotes the product of the non-zero eigenvalues of a matrix $A$ (see, e.g., \[b\].GodRoy/, Lemma 13.2.4). Thus, the asymptotic complexity of a sequence of graphs is a limit of the logarithm of the determinant of the graph Laplacians, appropriately normalized. 