Let T be a tree. Prove that if T has a vertex of degree k, then T has at least k leaves.
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Let T be a tree. Prove that if T has a vertex of degree k, then T has at least k leaves.
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- 1a. Prove: every tree with n ≥ 2 vertices has at least 2 leaves. (3 pt) 1b. Let T be a tree. Prove: if all vertices have degree either 1 or at least 4, then T has at least 2(n + 1)/3 leaves. (4 pt)Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a common vertex.A tournament is a digraph whose underlying graph is a complete graph. A root of a digraph is a vertex from which every vertex is reachable. A king of a digraph is a vertex u such that d(u,v)2 for every vertex v. Prove that every tournament has a root. Prove that every tournament has a king.
- Let T be a tree with p vertices of degree 1 and q other vertices. Show that the sum of the degrees of the vertices of degree greater than 1 is p+2(q-1).Let T be a rooted tree that contains vertices u, v, and w (among possibly others). Prove that if w is a descendant of both u and v, then u is a descendant of v or v is a descendant of u.Essentials of DISCRETE MATHEMATICS
- Let T be a tree of order n and suppose that all vertices of T have degree 1 or degree 3. Prove that T contains exactly n-2/2 vertices of degree 3Question 2 either 1 or 3. Let n = |V(T)|. Consider a tree T in which the degree of each vertex is (a) Show that n is even. (b) Show that T has 2 + 1 leaves. (c) Determine the number of distinct graphs G such that T is a spanning tree of G. Explain your reasoning.1 Let G = (V, E) be a connected graph that has two distinct spanning trees. Prove that |E| > |V] – 1.