Explanation Given information: Let G be a graph with n vertices, and let v and w be distinct vertices of G. Proof: The adjacency matrix A = [ a i j ] is n × n zero-one matrix with a i j = { 1 if there is an edge from v i to v j 0 otherwise A graph s connected if there exists a path between every pair of vertices. G is a graph with n vertices v and w are distinct vertices of G . To proof: If there is a walk from v to w, then there is a walk from v to w that has length less than or equal to n − 1. PROOF: Let us assume that there exists a walk W from v to w and let the length of this walk be m. If m ≤ n − 1 , then W is a walk of length less than or equal to n − 1 and thus then the proof would be complete To determine (b) Prove that G is connected if, and only if, every entry of A+A 2 + ... + A n − 1 is positive.

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ISBN: 9781337694193

Author: EPP, Susanna S.

Publisher: Cengage Learning,

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Chapter 10.2, Problem 23ES

To determine

(a)

Prove that if there is a walk from v to w, then there is a walk from v to w that has length less than or equal to n − 1.

To determine

(b)

Prove that G is connected if, and only if, every entry of $A+A2+...+An−1$ is positive.

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Chapter 10.2, Problem 22ES

Chapter 10.3, Problem 1TY

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Chapter 10 Solutions

Discrete Mathematics With Applications

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(a) Prove that if there is a walk from v to w, then there is a walk from v to w that has length less than or equal to n − 1.

Solution Summary: The author demonstrates that if there is a walk from v to w, then there are n vertices in the walk W. The adjacency matrix

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(a) Prove that if there is a walk from v to w, then there is a walk from v to w that has length less than or equal to n − 1.

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Chapter 10 Solutions

(a)

Prove that if there is a walk from v to w, then there is a walk from v to w that has length less than or equal to n − 1.