7. Show that a graph with n vertices is bipartite if, and only if, for some labeling ofits vertices, its adjacency matrix has the formAA'where A is a k × (n – k) matrix for some integerk such that 0 < k< n, the top left O representsa k× k matrix all of whose entries are 0, Ais the transpose of A, and the bottom right Orepresents an (n– k) × (n– k) matrix all ofwhose entries are 0.|-|

Show that a graph with n vertices is bipartite if, and only if, for some labeling of its vertices, its adjacency matrix has the form A A' where A is a k × (n– k) matrix for some integer k such that 0

Show that a graph with n vertices is bipartite if, and only if, for some labeling of its vertices, its adjacency matrix has the form A A' where A is a k × (n– k) matrix for some integer k such that 0

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 69EQ

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A: The answers are given below.

Question

7. Show that a graph with n vertices is bipartite if, and only if, for some labeling of
its vertices, its adjacency matrix has the form
A
A'
where A is a k × (n – k) matrix for some integer
k such that 0 < k< n, the top left O represents
a k× k matrix all of whose entries are 0, A
is the transpose of A, and the bottom right O
represents an (n– k) × (n– k) matrix all of
whose entries are 0.
|
-
|

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