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