Let T be a spanning tree of Kn, the complete graph on n vertices. What is the largest possible distance in T between two of its nodes? Show your completed work.
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A: The question has been answered in step2
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- Assume you have been given a graph with 1 minimum spanning tree and no repeated edge weights. Using the min-cut theorem, prove Boruvka's algorithm is correct. The pseudocode is attached.As for the following graph, find out all spanning trees. Among them, find out the (a) minımum spanning tree. 12Let G be a simple connected graph with 23 vertices and 24 edges. Compute the largest number of spanning trees that G can have. Justify your answer.
- Please written by computer sourceLet G = (V, E) be a connected graph that has two distinct spanning trees. Prove that |E| > |V] – 1.Let G be a graph, where each edge has a weight. A spanning tree is a set of edges that connects all the vertices together, so that there exists a path between any pair of vertices in the graph. A minimum-weight spanning tree is a spanning tree whose sum of edge weights is as small as possible. Last week we saw how Kruskal's Algorithm can be applied to any graph to generate a minimum-weight spanning tree. In this question, you will apply Prim's Algorithm on the graph below. You must start with vertex A. H 4 4 1 3 J 2 C 10 4 8 B 9 F 18 8 There are nine edges in the spanning tree produced by Prim's Algorithm, including AB, BC, and IJ. Determine the exact order in which these nine edges are added to form the minimum-weight spanning tree. 3.
- Let G (V, E) be a digraph in which every vertex is a source, or a sink, or both a sink and a source. (a) Prove that G has neither self-loops nor anti-parallel edges.Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace eachedge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.Let G be a graph with V vertices and E edges. One can implement Kruskal's Algorithm to run in O(E log V) time, and Prim's Algorithm to run in O(E + V log V) time. If G is a dense graph with an extremely large number of vertices, determine which algorithm would output the minimum-weight spanning tree more quickly. Clearly justify your answer.
- Hey, Kruskal's algorithm can return different spanning trees for the input Graph G.Show that for every minimal spanning tree T of G, there is an execution of the algorithm that gives T as a result. How can i do that? Thank you in advance!For Complete K₂, n-cube №n, and complete bipartite graph Km.n . a) How many vertices and how many edges do Qn, K and Km,n have? b) What is the degree of each vertex in Qn, Kn and Km,n?