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Chapter 10: Problem 6

Suppose that \(\mathcal{G}\) is a graph with adjacency matrix \(A\). (a) [BB] Show that the number of walks of length 2 in \(\mathcal{G}\) is the sum of the entries of the matrix \(A^{2}\). (b) Let \(d_{i}\) denote the degree of the vertex \(v_{i}\) in \(\mathcal{G}\). Show that the sum of the entries of \(A^{2}\) is also \(\sum d_{i}^{2}\).

Short Answer

Expert verified
The sum of entries in \( A^2 \) equals \( \sum d_i^2 \), both counting two-step walks.

Step by step solution

01

Understanding Walks in Graphs

A walk of length 2 in a graph \( \mathcal{G} \) is a sequence of vertices \( (v_i, v_j, v_k) \) such that \( v_i \to v_j \to v_k \) are edges in the graph. Thus, it involves two edges connecting three vertices.
02

Multiplying the Adjacency Matrix

Compute the square of the adjacency matrix, \( A^2 \). For a graph with adjacency matrix \( A \), the entry \( (i,j) \) in \( A^2 \) represents the number of two-step walks from vertex \( v_i \) to vertex \( v_j \). This is because \( (A^2)_{ij} = \sum_{k} A_{ik}A_{kj} \), summing over all possible intermediate vertices \( v_k \).
03

Summing the Entries of \( A^2 \)

The sum of all entries in \( A^2 \), \( \sum_{i,j} (A^2)_{ij} \), gives the total number of walks of length 2 in the graph. This is because each entry counts the walks between specific pairs \( (v_i, v_j) \), and their sum covers all possible pairs.
04

Vertex Degree in Graphs

The degree \( d_i \) of a vertex \( v_i \) is the number of edges incident to \( v_i \), which is equivalent to the sum of entries in the \( i \)-th row (or column, in an undirected graph) of the adjacency matrix, i.e., \( d_i = \sum_j A_{ij} \).
05

Calculating \( \sum d_i^2 \)

Calculate \( \sum d_i^2 = \sum (\sum_j A_{ij})^2 \). Expanding this square gives \( \sum_i \sum_{j,k} A_{ij}A_{ik} \). This is equivalent to \( \sum_{i,j,k} A_{ij}A_{ik} = \text{sum of all entries in } A^2 \), as each term represents two-edge walks in the graph.
06

Conclusion

From Steps 3 and 5, we show that both expressions, the sum of entries of \( A^2 \) and \( \sum d_i^2 \), are equivalent because they both count the total number of two-step walks in the graph.

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Key Concepts

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

Adjacency Matrix
The adjacency matrix of a graph is a square matrix used to represent the connections between vertices in the graph. Each row and column in the matrix corresponds to a vertex in the graph.
For a graph with vertices numbered 1 to n, if there is an edge between vertices i and j, the entry (i, j) in the matrix is 1; otherwise, it is 0. This means the matrix is filled with 0s and 1s.
The adjacency matrix is powerful because you can use it for algebraic operations to infer different properties of the graph.
  • It allows you to determine whether there is a direct connection between two vertices.
  • It helps in counting specific paths or walks by utilizing its powers, which we will delve into in the graph powers section.
In general, understanding the adjacency matrix is crucial for both visual and computational graph analysis.
Walks in Graphs
A walk in a graph is a sequence of vertices where each consecutive pair is connected by an edge. Walks can vary in length, representing different paths through the graph.
For example, a walk of length 2 involves exactly three vertices connected by two edges, taking the form (v_i, v_j, v_k).
Walks are important because they allow us to explore and analyze the connectivity within a graph.
  • They are a sequence of vertices with certain permissions: each step (or edge) must lead to another vertex in the graph.
  • Walks reflect how information or resources can flow through a network represented by a graph.
Understanding walks in graphs helps in understanding complex network flow, such as communication networks and social networks.
Vertex Degree
The degree of a vertex in a graph is the number of edges connected to it. Essentially, it counts how many other vertices are directly connected to a given vertex.
This concept is crucial because it gives insight into the importance or influence of a vertex in a network.

Mathematically, the degree of a vertex can be expressed as the sum of a row (or column, as graphs are often symmetrical) in the adjacency matrix:
  • For vertex v_i, the degree is denoted as d_i = \( \sum_j A_{ij} \).
  • In graphs, high-degree vertices can represent hubs or central points of interaction.
  • The degree helps in identifying connectivity and possible weak points within a network.
Calculating the degrees of vertices is a foundational step in analyzing graphs and their properties.
Graph Powers
Graph powers offer deeper insights into the connections within a graph by representing multiplied paths. When you square the adjacency matrix, it reflects walks of length 2 between vertices.
For instance, in an adjacency matrix A, multiplying it by itself to form A虏, each entry (i,j) shows the number of two-step walks between vertex v_i and vertex v_j.
  • This multiplication considers all intermediate vertices that facilitate the walk.
  • Efficient in showcasing patterns within the graph structure, such as potential clusters.
  • Summing all the entries of A虏 gives the total number of two-step walks across the entire graph.
The concept of graph powers is a gateway to understanding more complex interactions in a network beyond direct connections, like predicting pathways and evaluating structural robustness.

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Most popular questions from this chapter

Suppose \(A_{1}\) and \(A_{2}\) are the adjacency matrices of isomorphic graphs \(\mathcal{G}_{1}\) and \(\mathcal{G}_{2}\), respectively. Show that \(A_{1}\) and \(A_{2}\) have the same characteristic polynomial. [Hint: First show that if \(P\) is a permutation matrix, then \(P^{T}=\) \(\left.P^{-1} .\right]\)

Draw a picture of a cube. By imagining that the bottom square is stretched until it is flat, draw a graph of the flattened cube. Is this graph Hamiltonian? If so, draw a Hamiltonian cycle. If not, explain why not.

What is the adjacency matrix of \(\mathcal{K}_{n} ?\) Label the vertices of \(\mathcal{K}_{m, n}\) so that the adjacency matrix has an especially nice form.

Inadvertently, David reverses the order of the \(k\) and the i, \(j\) loops in Step 2 of the Floyd-Warshall algorithm. implementing Step 2 in the form For \(i, j=1\) to \(n\) for \(k=1\) to \(n\), let \(d(i, j)=\min \\{d(i, j), d(i, k)+\) \(d(k, j)\\}\) Does this affect the algorithm? Explain.

For each pair of matrices \(A_{1}, A_{2}\) shown, decide whether or not there is a permutation matrix \(P\) with \(A_{2}=\) \(P A_{1} P^{T}\). Either find \(P\) or explain why no such \(P\) exists. (a) \([\mathrm{BB}] A_{1}=\left[\begin{array}{llll}0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0\end{array}\right]\), \(A_{2}=\left[\begin{array}{llll}0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0\end{array}\right]\) (b) \(A_{1}=\left[\begin{array}{lllll}0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & 0 \\\ 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0\end{array}\right]\) \(A_{2}=\left[\begin{array}{lllll}0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0\end{array}\right]\) (c) \(A_{1}=\left[\begin{array}{llllll}0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0\end{array}\right]\), \(A_{2}=\left[\begin{array}{llllll}0 & 1 & 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 & 0\end{array}\right]\)
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