A leaf in a tree T is a vertex x E V(T) of degree 1. (a) Show that every finite tree T has at least A(T) leaves. (b) Deduce that every finite tree T with |V(T)| >= 2 has at least two leaves.

Suppose that an airplane flies along the path r₁(t) = (2t, 2t², 10e¯¹) and a skydiver takes the path r₂(t) = (2,2, 10-2t), for 0≤ t ≤ 5. Do their paths cross? Do they collide?

Compute the convolution x[n]*h[n] for all n > 0, by using Graphical or Tabular method. x[0] = 3, x[1] = 5, x[2] = 7, x[3] = 6, x[4] = 4, x[n] = 0 for all other integers (n). h[0] = 5, h[1] = -1, h[2] = 2, h[3] = 4, h[n] = for all other integers (n). • y[n]=x[n]*h[n] • v[n]= Ž x[* I» [n – k] *--00

4.1.12. Let n, k be positive integers with n even, k odd, and n > k > 1. Let G be the k- regular simple graph formed by placing n vertices on a circle and making each vertex adjacent to the opposite vertex and to the (k – 1)/2 nearest vertices in each direction. Prove that < (G) = k. (Harary (1962a])