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Prove that any graph has at least two vertices with the same degree. A complete bipartite graph on ( m , n ) vertices, is a simple graph whose vertices can be divided into two distinct, non-overlapping sets (that is, suppose V has m vertices and W has n vertices) in such a way that there is exactly one edge from each vertex of V to each vertex of W , there is no edge from any one vertex of V to any other vertex of V , and there is no edge from any one vertex of W to any other vertex of W. Use ways to select the edges to show that this graph has m x n edges Use combinations to show that the number of edges on a complete graph is n(n-1)/2

Prove that any graph has at least two vertices with the same degree. A complete bipartite graph on ( m , n ) vertices, is a simple graph whose vertices can be divided into two distinct, non-overlapping sets (that is, suppose V has m vertices and W has n vertices) in such a way that there is exactly one edge from each vertex of V to each vertex of W , there is no edge from any one vertex of V to any other vertex of V , and there is no edge from any one vertex of W to any other vertex of W. Use ways to select the edges to show that this graph has m x n edges Use combinations to show that the number of edges on a complete graph is n(n-1)/2

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 80EQ
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2. Prove that any graph has at least two vertices with the same degree. A complete bipartite graph on ( m, n ) vertices, is a simple graph whose vertices can be divided into two distinct, non-overlapping sets (that is, suppose V has m vertices and W has n vertices) in such a way that there is exactly one edge from each vertex of V to each vertex of W, there is no edge from any one vertex of V to any other vertex of V , and there is no edge from any one vertex of w to any other vertex of W. Use ways to select the edges to show that this graph has m x n edges Use combinations to show that the number of edges on a complete graph is n(n-1)/2 (NOTE: Please elaborate on the answer and explain. Please do not copy-paste the answer from the internet or from Chegg.)
Transcribed Image Text:2. Prove that any graph has at least two vertices with the same degree. A complete bipartite graph on ( m, n ) vertices, is a simple graph whose vertices can be divided into two distinct, non-overlapping sets (that is, suppose V has m vertices and W has n vertices) in such a way that there is exactly one edge from each vertex of V to each vertex of W, there is no edge from any one vertex of V to any other vertex of V , and there is no edge from any one vertex of w to any other vertex of W. Use ways to select the edges to show that this graph has m x n edges Use combinations to show that the number of edges on a complete graph is n(n-1)/2 (NOTE: Please elaborate on the answer and explain. Please do not copy-paste the answer from the internet or from Chegg.)
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