Solution
= Solution
$(A^r)_{ij}$ counts length-$r$ walks from $i$ to $j$, so $\operatorname{tr}A^r$ counts length-$r$ closed walks. The given power sums identify the <spectrum> as $5$ once, $\sqrt5$ and $-\sqrt5$ three times each, and $-1$ five times. Thus $n=12$, regular degree $d=5$, and the size is $nd/2=30$.