The spectral moments of trees with given maximum degree

Let Δ ≥ 3 . Denote by T n , Δ the set of all trees with n vertices and maximum degree Δ and by T n , Δ ∗ the set of all Δ -trees with n vertices. In this work, we first show that all Δ -trees come before all trees in T n , Δ ∖ T n , Δ ∗ in an S -order and present a criterion for a Δ -tree coming before another Δ -tree in an S -order. Also, we give the first ∑ k = 1 ⌊ n − 1 3 ⌋ ( ⌊ n − k − 1 2 ⌋ − k + 1 ) graphs apart from a path, in an S -order, of all trees with n vertices.