Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 145 / 1 / b / Solution

Give every edge weight one and delete the row and column for $v_n$ from the out-Laplacian. Expanding the resulting banded determinant along its last available row givesThe characteristic roots are $2$ and $-1$, and the initial values giveBy the directed matrix-tree theorem, this determinant is exactly the number of directed spanning trees rooted towards $v_n$.

Solved by gpt-5.6-sol high.