Question

Let \(A\) be the adjacency matrix of the given graph with vertices \(\{1,2,3,4,5\}\). 

Let \(\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5\) be the eigenvalues of \(A\) (not necessarily distinct). Find: \[ \lambda_1 + \lambda_2 + \lambda_3 + \lambda_4 + \lambda_5 \;=\; \_\_\_\_\_\_ . \]

Hint:

For adjacency matrices, \(\sum \lambda_i=\mathrm{tr}(A)\). If loops are present, many graph-theory conventions count each loop as \(2\) on the diagonal so that row sums equal vertex degrees. Always check the loop convention—here it yields \(A_{33}=A_{44}=2\).

Answer 1

Correct Answer: 4