Let $A$ be the adjacency matrix of the graph with vertices $\{1, 2, 3, 4, 5\}$. Let $\lambda_1, \lambda_2, \lambda_3, \lambda_4,$ and $\lambda_5$ be the five eigenvalues of $A$. Note that these eigenvalues need not be distinct. The value of $\lambda_1 + \lambda_2 + \lambda_3 + \lambda_4 + \lambda_5 =$ __________.

The sum of all eigenvalues of a square matrix is equal to its trace (the sum of its main diagonal elements):

In the adjacency matrix of a graph:

• A diagonal entry $A_{ii}$ represents the number of self-loops at vertex $i$.

• Vertices $1, 2,$ and $5$ have no self-loops, so $A_{11} = A_{22} = A_{55} = 0$.

• Vertices $3$ and $4$ each have one self-loop.

Depending on the convention used for self-loops in an adjacency matrix:

1. Standard convention: A self-loop contributes $1$ to the diagonal entry. Thus, $A_{33} = 1$ and $A_{44} = 1$.

   $$\text{Trace}(A) = 0 + 0 + 1 + 1 + 0 = 2$$

2. Degree-preserving convention: A self-loop contributes $2$ to the diagonal entry. Thus, $A_{33} = 2$ and $A_{44} = 2$.

   $$\text{Trace}(A) = 0 + 0 + 2 + 2 + 0 = 4$$

Both 2 and 4 are accepted by the official answer key [2 to 2 OR 4 to 4].