On a poset of trees

We will prove that the path minimizes the number of closed walks of length ℓ among the connected graphs for all ℓ. Indeed, we will prove that the number of closed walks of length ℓ and many other properties such as the spectral radius, Estada index increase or decrease along a certain poset of trees. This poset is a leveled poset with path as the smallest element and star as the greatest element.