Let A be the adjacency matrix of the graph with vertices {1, 2, 3, 4, 5}. Let…
2023
Let A be the adjacency matrix of the graph with vertices {1, 2, 3, 4, 5}.
Let λ1, λ2, λ3, λ4, and λ5 be the five eigenvalues of A. Note that these eigenvalues need not be distinct. The value of λ1 + λ2 + λ3 + λ4 + λ5 = _______.
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Correct answer: 2
Key fact: the sum of the eigenvalues of a matrix equals its trace (the sum of its diagonal entries).
In an adjacency matrix, a 1 on the diagonal indicates a loop at that vertex. In the given graph vertices 3 and 4 each have a loop, so a_33 = 1 and a_44 = 1; all other diagonal entries are 0.
Thus trace(A) = 1 + 1 = 2, so the sum of the five eigenvalues is 2.