2021-048

Consider a random network such that \( N = 100 \) and \( \langle k \rangle = 0.5 \). What is the standard deviation of the network's degree distribution (round to two decimal places)?

1. 0.50
2. 0.58
3. 0.69
4. 0.71
5. None of the above

Original idea by: Helena Vieira

2021-047

For each of the following graphs determine whether it is planar, bipartite, both, or none.

(I)

(II)

(III)

(IV)

A. I-Bipartite, II-None, III-Planar, IV-Bipartite

B. I-Planar and Bipartite, II-None, III-Planar, IV-Bipartite

C. I-Planar and Bipartite, II-Planar and Bipartite, III-Bipartite, IV-None

D. I-Planar, II-Planar and Bipartite, III-None, IV-Planar

E. None of the above

 

 Original idea by: Taylla Milena Theodoro

2021-046

 

Consider the following graph:

 

The average degree and average clustering coefficient values for this graph are, respectively:

A. 16/7 and 4/7  

B. 16/7 and 3/7

C. 15/7 and 4/7

D. 11/3 and 1/3

E. None of the above

2021-045

Let \( \bar{G} \) denote the complement graph of an undirected, loppless graph \( G = (V, E) \), that is, \( \bar{G} \) has the same set of vertices \( V \), but the set of edges in \( \bar{G} \) consists of every pair of vertices from \( V \) that is not included in \( E \).

How can we express the number of edges in \( \bar{G} \) in terms of the sizes \( v = |V| \) and \( e = |E| \) ?

1. $$ \frac{2v^2 + 2v + e}{2} $$
2. $$ \frac{v^2 - v - 2e}{2} $$
3. $$ \frac{2v^2 - v + 2e}{2} $$
4. $$ \frac{v^2 + 2v - e}{2} $$
5. None of the above

Original idea by: Jader Martins

2021-044

Consider that you are a postman and have been asked to deliver some packages in different neighborhoods in two cities: Sorocaba and Campinas. For this, the logistics company provided a diagram that represents the possible routes. As you are an excellent student of Network Science, you remember that it is possible to build an Adjacency Matrix to help you in this choice. Which alternative represents these matrices, assuming the order of nodes in the rows and columns is 1,2,3,4,5,6 for Sorocaba and A,B,C,D,E,F for Campinas?

Sorocaba and Campinas Neighborhoods

1. \( \left( \begin{array}{cccccc} 0&0&1&0&0&0 \\ 0&0&1&1&0&0 \\ 1&1&0&0&1&1 \\ 0&1&0&0&0&0 \\ 0&0&1&0&0&0 \\ 0&0&1&0&0&0 \end{array} \right) \) , \( \left( \begin{array}{cccccc} 0&0&0&0&0&1 \\ 0&0&1&0&1&1 \\ 0&1&0&1&1&0 \\ 0&0&1&0&0&0 \\ 0&1&1&0&0&0 \\ 1&1&0&0&0&0 \end{array} \right) \)
2. \( \left( \begin{array}{ccccc} 0&0&1&0&0 \\ 0&0&1&1&0 \\ 1&1&0&0&1 \\ 0&1&0&0&0 \\ 0&0&1&0&0 \end{array} \right) \) , \( \left( \begin{array}{cccccc} 0&0&0&0&1 \\ 0&0&1&0&1 \\ 0&1&0&1&0 \\ 0&0&1&0&0 \\ 1&1&0&0&0 \end{array} \right) \)
3. \( \left( \begin{array}{ccccccc} 0&0&1&0&0&0 \\ 0&0&0&0&0&A \\ 0&0&2&1&0&0 \\ 0&0&B&0&1&1 \\ 1&1&0&0&5&1 \\ 0&1&0&A&1&0 \end{array} \right) \) , \( \left( \begin{array}{ccccccc} 0&3&0&0&0&0 \\ 0&0&1&0&0&0 \\ 0&0&4&0&0&0 \\ 0&C&D&0&0&0 \\ 0&0&1&0&6&0 \\ 1&1&0&0&E&0 \end{array} \right) \)
4. \( \left( \begin{array}{ccccccc} 1&0&1&0&0&0 \\ 0&1&0&0&0&1 \\ 1&0&1&1&0&0 \\ 0&1&1&0&1&1 \\ 1&1&0&0&1&1 \\ 0&1&0&1&1&0 \end{array} \right) \) , \( \left( \begin{array}{ccccccc} 0&1&0&0&0&0 \\ 0&0&1&1&0&0 \\ 0&0&1&0&0&0 \\ 0&1&1&0&0&0 \\ 1&0&1&0&0&0 \\ 1&1&0&1&0&0 \end{array} \right) \)
5. None of the above

Original idea by: Adson N Alves

2021-043

Choose the alternative that provides a solution for the following differential equation (assume that C is a constant):

$$ y' - 2 x y = x $$

1. \( 2 + 4x^2 \)
2. \( 4x^2 - x^3 \)
3. \( C e^{x^2} - 1/2 \)
4. \( C e^{3x} + 2 \)
5. None of the above

Original idea by: Thales Almeida

2021-042

Consider a function \( f(x) \) given by:

$$ f(x) = - x^\alpha \cos x + C x^\alpha . $$

Choose the value of \( \alpha \) that makes the following formula true.

$$ x f' - 2 f = x^3 \sin x $$

1. \( \alpha = 0 \)
2. \( \alpha = 1 \)
3. \( \alpha = 2 \)
4. \( \alpha = 3 \)
5. None of the above.

Original idea by: Thales Almeida