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Chapter 7: Problem 10

Find the adjacency list representation of the relation with the given adjacency matrix.

Short Answer

Expert verified
The adjacency list representation for the given relation with the adjacency matrix: \[ \begin{bmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{bmatrix}\] is as follows: 1: [2] 2: [1, 3, 4] 3: [2] 4: [2]

Step by step solution

01

Understanding adjacency list and adjacency matrix

An adjacency matrix is a square matrix that is used to represent a finite graph. It indicates the presence of a relation between elements using '1' and absence using '0'. An adjacency list is an alternative representation of a graph where each node has a list of its adjacent nodes. Step 2: Analyzing adjacency matrix
02

Analyzing adjacency matrix

Go through the adjacency matrix and identify the elements with value '1'. These will represent the pairs of nodes that are connected. Here is an overview of the connections from the matrix: (\(1 \rightarrow 2\)), (\(2 \rightarrow 1\)), (\(2 \rightarrow 3\)), (\(2 \rightarrow 4\)), (\(3 \rightarrow 2\)), (\(4 \rightarrow 2\)) Step 3: Creating adjacency list representation
03

Creating adjacency list representation

Now that we have identified the connections, create the adjacency list representation. Each entry should be a list of nodes directly connected to the current node. Node 1: [2] Node 2: [1, 3, 4] Node 3: [2] Node 4: [2] The adjacency list representation for the given relation with the adjacency matrix is: 1: [2] 2: [1, 3, 4] 3: [2] 4: [2]

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Most popular questions from this chapter

Let \(R\) and \(S\) be relations on a set. Prove each. Let \(A, B, C,\) and \(D\) be any sets, \(R\) a relation from \(A\) to \(B, S\) a relation from \(B\) to \(C,\) and \(T\) a relation from \(C\) to \(D .\) Prove that \(R \odot(S \odot T)=(R \odot S) \odot T\) (associative property)

Write an algorithm to find each. The \(n\) th boolean power of an \(m \times m\) boolean matrix \(A\)

Find the partition of the set \(\\{a, b, c\\}\) induced by each equivalence relation. $$\\{(a, a),(a, c),(b, b),(c, a),(c, c)\\}$$

Libraries use a sophisticated code-a-bar system to assign each book a 13 -digit identification number \(d_{1}, d_{2} \ldots d_{13}\) and a check digit d. Let \(k\) denote the number of digits among \(d_{1}, d_{3}, d_{5}, d_{7}, d_{9}\) \(d_{11},\) and \(d_{13}\) greater than or equal to \(5 .\) Then \(d\) is computed as \(\left.d \equiv 1 \left(d_{1}, d_{2}, \ldots, d_{13}\right) \cdot(2,1,2,1,2,1,2,1,2,1,2,1,2)-k\right] \bmod 10\) where the dot indicates the dot product. Compute the check digit for 2,035,798,008,938

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