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Chapter 4: Problem 34

Prove that a complete graph (a graph in which there is an edge between every pair of vertices has \(n^{n-2}\) spanning trees, Here \(n\) is the number of vertices in the graph.

Short Answer

Expert verified
There are \(n^{n-2}\) distinct labelled trees or spanning trees in a complete graph with \(n\) vertices.

Step by step solution

01

Understanding the given

In a complete graph, there exists an edge between every pair of vertices. A Spanning Tree of a graph is a tree formed using the graph's vertices and a subset of the graph's edges. Our goal is to prove the relation between the number of vertices and the number of spanning trees in a complete graph, where the number of spanning trees is given by \(n^{n-2}\).
02

Considering the Prüfer Code

Prüfer code is a one-to-one correspondence between the set of labeled trees (trees where the vertices are distinct) with \(n\) vertices and the set of sequences with \(n-2\) elements. The code is constructed by removing vertices from the graph and noting the vertex of highest order that is connected to the removed vertex by an edge.
03

Proof Explanation

A complete graph with \(n\) vertices will have \((n-2)\) elements in the Prüfer code sequence, and each element can be any of the \(n\) vertices. As there is a one-to-one correspondence between Prüfer codes and spanning trees, there are \(n^{(n-2)}\) possible Prüfer sequences. Consequently, there are \(n^{(n-2)}\) distinct labelled trees or spanning trees in a complete graph with \(n\) vertices.

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Most popular questions from this chapter

Assume that in a network of computers any two computers can be linked. Given a cost estimate for each possible link, should Algorithm 4.1 (Prim's Algorithm) or Algorithm 4.2 (Kruskal's Algorithm) be used? Justify your answer.

Can Dijkstra's Algorithm (Algorithm 4.3 ) be used to find the shortest paths in a graph with some negative weights? Justify your answer.

Use induction to prove the correctness of Dijkstra's Algorithm (Algorithm 4.3).

Suppose we assign \(n\) persons to \(n\) jobs. Let \(C_{U}\) be the cost of assigning the ith person to the jth job. Use a greedy approach to write an algorithm that finds an assignment that minimizes the total cost of assigning all \(n\) persons to all \(n\) jobs. Analyze your algorithm, and show the results using order notation.

Use a greedy approach to write an algorithm for the Traveling Salesperson Problem. Show that your algorithm does not always find a minimum-length tour.
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