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Chapter 3: Problem 60

(a) Prove that a graph is bipartite if and only if its vertices can be labeled so that its adjacency matrix can be partitioned as \\[ A=\left[\begin{array}{ll} O & B \\ B^{T} & O \end{array}\right] \\] (b) Using the result in part (a), prove that a bipartite graph has no circuits of odd length.

Short Answer

Expert verified
A bipartite graph's adjacency matrix can be partitioned into a block form, ensuring no odd-length circuits.

Step by step solution

01

Understanding Bipartite Graphs

A graph is bipartite if we can divide its set of vertices into two disjoint sets such that no two vertices within the same set are adjacent. This means there is no edge connecting vertices of the same set.
02

Defining the Adjacency Matrix

For a graph with n vertices, the adjacency matrix is an n x n matrix A where the entry \(A_{ij}\) is 1 if there is an edge between vertices i and j, and 0 otherwise. For a bipartite graph, the vertices can be split into two sets such that there are no edges within the same set.
03

Constructing the Block Matrix

To align with the block matrix \[A=\left[\begin{array}{cc}O & B \B^{T} & O \\end{array}\right]\], define vertices V = \(V_1 \cup V_2\), where V1 and V2 are the two parts of the bipartite graph. The top left and bottom right blocks, represented by O, are zero matrices, illustrating there are no connections within the same part.
04

Analyzing B and \(B^T\)

Block B represents the connections from V1 to V2, while \(B^T\) is the transpose of B, capturing edges from V2 to V1. Since these blocks contain connections only between the two separate sets, a bipartite graph adjacency matrix fits this block form.
05

Proving No Odd-Length Circuits

An odd-length circuit requires returning to the starting vertex after an odd number of moves. However, in a bipartite graph, if you start on one side (say V1), after an odd number of moves, you must end up on the opposite side (V2), not back on the starting vertex, since after an even number of moves (2, 4, etc.), one returns to the start.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Adjacency Matrix
An adjacency matrix is an essential tool in graph theory. It represents connections between vertices in a graph using a square matrix format. The size of the matrix is determined by the number of vertices in the graph. Each entry in the matrix indicates whether a pair of vertices are connected by an edge.
  • If the entry \(A_{ij}\) is 1, there is an edge between vertex \(i\) and vertex \(j\).
  • If the entry \(A_{ij}\) is 0, no edge exists between the vertices.
For a bipartite graph, the adjacency matrix can be cleverly arranged. By partitioning it, the adjacency matrix can be represented as a block matrix form that showcases connections only between two different sets of vertices, ensuring no internal set connections.
Block Matrix
A block matrix is a matrix divided into submatrices or "blocks". This structure allows different sections of the matrix to be manipulated separately.
In the context of bipartite graphs, an adjacency matrix can take the form of a block matrix as follows:\[A=\left[\begin{array}{cc}O & B \B^{T} & O\end{array}\right]\]
  • The "O" blocks are zero matrices, indicating no connections within each vertex set.
  • The "B" block contains data about edges from the first vertex set (\(V_1\)) to the second (\(V_2\)).
  • The "B^T" block shows the transpose of B, representing edges from \(V_2\) back to \(V_1\).
This structure clearly demonstrates that in a bipartite graph, edges only exist between different sets and not within the same set.
Circuit
A circuit in graph theory refers to a path that starts and ends at the same vertex without revisiting any vertex along the way. It's a closed loop within a graph.
In bipartite graphs, circuits of odd length are impossible. This is because:
  • A bipartite graph consists of two disjoint sets of vertices (\(V_1\) and \(V_2\)).
  • Any path that starts in \(V_1\) might end back in \(V_1\) only after an even number of moves (i.e., 2, 4, 6, etc.).
  • After an odd number of moves, the path would end in the opposite set \(V_2\), not forming a closed loop.
This property is a fundamental characteristic of bipartite graphs, ensuring they do not have circuits of odd length.
Graph Theory
Graph theory is a branch of mathematics focusing on the study of graphs, which are structures used to model pairwise relations between objects.
Graphs are fundamental in understanding complex networks in various fields such as computer science, social sciences, and biology.
A bipartite graph is a special kind of graph in graph theory, characterized by the partitioning of its vertex set into two disjoint subsets where no two graph vertices within the same subset are adjacent.
  • The understanding of adjacency and block matrices aids in determining if a graph is bipartite.
  • The concept of circuits and their lengths helps to analyze the connectivity and properties of graphs at a deeper level.
Graph theory provides tools to model and solve real-world problems concerning networks and connections efficiently.

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Most popular questions from this chapter

Determine whether b is in \(\operatorname{col}(A)\) and whether \(\mathbf{w}\) is in row\((A),\) as in Example 3.41. $$A=\left[\begin{array}{rrr} 1 & 1 & -3 \\ 0 & 2 & 1 \\ 1 & -1 & -4 \end{array}\right], \mathbf{b}=\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right], \mathbf{w}=\left[\begin{array}{lll} 2 & 4 & -5 \end{array}\right]$$

Compute the rank and nullity of the given matrices over the indicated \(\mathbb{Z}_{p}\). $$\left[\begin{array}{llll} 1 & 3 & 1 & 4 \\ 2 & 3 & 0 & 1 \\ 1 & 0 & 4 & 0 \end{array}\right] \text { over } \mathbb{Z}_{5}$$

A population with three age classes has a Leslie matrix \(L=\left[\begin{array}{lll}1 & 1 & 3 \\ 0.7 & 0 & 0 \\ 0 & 0.5 & 0\end{array}\right] .\) If the initial population vector is \(\mathbf{x}_{0}=\left[\begin{array}{l}100 \\ 100 \\ 100\end{array}\right],\) compute \(\mathbf{x}_{1}, \mathbf{x}_{2},\) and \(\mathbf{x}_{3}\).

Verify Theorem 3.32 by finding the matrix of S o \(T\) (a) by direct substitution and (b) by matrix multiplication of \([S][T]\). $$T\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]=\left[\begin{array}{c} x_{2} \\ -x_{1} \end{array}\right], S\left[\begin{array}{l} y_{1} \\ y_{2} \end{array}\right]=\left[\begin{array}{c} y_{1}+3 y_{2} \\ 2 y_{1}+y_{2} \\ y_{1}-y_{2} \end{array}\right]$$

Answer by considering the matrix with the given vectors as its columns. Do \(\left[\begin{array}{r}1 \\ -1 \\ 0 \\\ 0\end{array}\right],\left[\begin{array}{r}0 \\ 1 \\ 0 \\\ -1\end{array}\right],\left[\begin{array}{r}0 \\ 0 \\ -1 \\\ 1\end{array}\right],\left[\begin{array}{r}-1 \\ 0 \\ 1 \\\ 0\end{array}\right]\) form a basis for \(\mathbb{R}^{4} ?\)
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