2026-314

Let \(A\) be the \(N \times N\) adjacency matrix of an undirected, unweighted network without self-loops. Let \( \textbf{1}\) be a column vector of \(N\) elements, all equal to 1, i.e., \( \textbf{1} = (1, 1, \ldots, 1)^T \). Let \(t(A)\) denote the trace of matrix \(A\), i.e., the sum of the elements on its main diagonal.

Which of the following expressions gives the number of connected triplets in the network?

1. \( ( \textbf{1}^T A^2 \textbf{1} ) / 2 \)
2. \( ( \textbf{1}^T A^2 \textbf{1} - t(A^2) ) / 2 \)
3. \( ( \textbf{1}^T A^2 \textbf{1} - t(A^2) - t(A^3) ) / 2 \)
4. \( ( \textbf{1}^T A^2 \textbf{1} - t(A^2) ) / 2 - t(A^3)/6 \)
5. None of the above

Original idea by: Pedro Ferreira

2026-313

 Consider the following networks:

Choose the correct answer:

A) Only networks A, B, and D are bipartite.

B) All networks are bipartite.

C) Only networks A, B, and C are bipartite.

D) Only networks A and D are bipartite.

E) None of the above.

Original idea by: Melissa Araújo

2026-312

Look at the figures below, analyze the statements, and choose the right answer:

I- The graph in figure A is a multigraph and in figure B is a simple graph.

II- In a complete graph, known as a clique, all nodes are connected to each other.
III- The diameter of the graph in figure B is dmax= 4.

IV- The adjacency matrix of graph in figure B is:

1. I and II are correct
2. II, III and IV are correct
3. II and IV are correct
4. I, III, IV are correct
5. None of the above

Original idea by: Tássia Martins

2026-311

 Consider the following network.

Which of the following statements are true:

I. FECFAU and FEAGRI are the only nodes with the lowest degree.

II. The degree distribution shows that 60% of the nodes have degree 3.

III. The local clustering coefficient of IQ and IE is 1.

IV. The diameter of the graph is 4.

V. No links are bridges

a) I, II, and III.

b) III and V.

c) I, III, and IV.

d) II, IV, and V.

e) None of the above.

Original idea by: George Gigilas Junior