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

Step 1: Use a basic property of matrices

For any square matrix, the sum of all eigenvalues is equal to the sum of its diagonal elements. Thus, adding the diagonal entries of the adjacency matrix is sufficient.

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Step 2: Interpret diagonal entries from the graph

In the adjacency matrix of a graph:

• A vertex without a self-loop contributes 0 on the diagonal
• A vertex with a self-loop contributes 2 on the diagonal (loop counted twice)

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Step 3: Identify looped vertices

From the given graph:

• Vertex 3 has a self-loop → diagonal contribution = 2
• Vertex 4 has a self-loop → diagonal contribution = 2
• Vertices 1, 2, and 5 have no self-loops → diagonal contribution = 0

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Step 4: Add diagonal contributions

0 + 0 + 2 + 2 + 0 = 4

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Final Answer:

4