Theorem 4.34 (Godsil). Assuming that:

  • G is d-regular

  • λ1≤⋯≤λn are the roots of μG(x)

Then
∑k=1nλkl=∑aWal(G),

where Wal(G) is the number of closed walks of length l from a in the path tree Ta(G) of G.