Quiz Question #1

Figure 1(a) below shows a drawing of a network \(N\). Figures 1(b), 1(c), and 1(d) shows the matrices \(\boldsymbol A\), \(\boldsymbol D\), \(\boldsymbol S\), respectively, and Figure 1(e) shows the vector \(\boldsymbol{k}\).

 

 

 

 

Regarding the figures above, consider the following six propositions:

1. For matrix \(\boldsymbol A\) (shown in Figure 1(b)) entry \(a_{ij} = 1\) if and only if there is a link in network \(N\) connecting node \(i\) to node \(j\). Therefore, \(\boldsymbol A\) is an adjacency matrix of network \(N\).
2. Matrix \(\boldsymbol D\) (shown in Figure 1(c)) can be regarded as a matrix indicating second neighbors (nodes at distance two) of network \(N\), since entry \(d_{ij} \neq \infty\) if and only if nodes \(i\) and \(j\) are second neighbors.    
3. Matrix \(\boldsymbol D\) (shown in Figure 1(c)) can be regarded as distance matrix of network \(N\) , since if  \(d_{ij} = \infty\) there is no path between nodes \(i\) and \(j\), and if \(d_{ij} \neq \infty\), then \(d_{ij}\) equals the length of a shortest path between nodes \(i\) and \(j\).
4. Matrix \(\boldsymbol S\) (shown in Figure 1(c)) can be regarded as a matrix of second neighbors of netwok \(N\), since \(s_{ij} =1\) if, and only if, nodes \(i\) and \(j\) are second neighbors.    
5. Vector \(\boldsymbol{k}\) is the vector of the node degrees of network \(N\) and can be obtained multiplying matrix \(A\) by vector \(\begin{pmatrix}1 & 1 & 1 & 1 & \cdots& 1\end{pmatrix}^T\).  

 

Which alternative contains all TRUE propositions?

(a)  1 and 3.

(b) 1, 2, and 3.

(c) 1, 2, and 5.

(d) 1, 3, 4, and 5.

(e) None of the above.