Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 17J / b / Solution

$(A^r{)}_{ij}$ counts length-$r$ walks from $i$ to $j$, so $\mathrm{tr}{A}^r$ counts length-$r$ closed walks. The given power sums identify the spectrum as $5$ once, $\sqrt{5}$ and $-\sqrt{5}$ three times each, and $-1$ five times. Thus $n=12$, regular degree $d=5$, and the size is $nd/2=30$.