Exercise 6 in chapter 9 of Algebraic Combinatorics by Stanley: Let $G$ be a finite graph on $p$ vertices with Laplacian matrix $L(G)$. Let $G'$ be obtained from $G$ by adding a new vertex $v$ and connecting it to each vertex of $G$ (so we have $p$ new edges). Express $\kappa(G')$ (the number of spanning trees of $G'$) in terms of the eigenvalues $\mu_1, \dots, \mu_p$ of $L(G)$.
Clearly, $L(G')$ is just $L(G)$ with a row of $-1$'s and column of $-1$'s below and to the right of the submatrix $L(G)$, respectively, with an additional diagonal element $p$ in the bottom right corner of the matrix, since the added vertex $v$ has degree $p$. Then, by the Matrix-Tree theorem, $\kappa(G') = \det(L(G)) = \mu_1 \dots \mu_p$. However, this inspired the question: if we had only connected $v$ with a select subset of the original $p$ matrices to get some new graph $H$, then we would still get that $\kappa(H) = \det(L(G)) = \kappa(G')$. Thus, even if $v$ was an isolated vertex, we would still have the same number of spanning trees, but this is clearly wrong, as no spanning trees would exist in this case.
