GATE CSE 2023 Q30: Sum of eigenvalues of a graph's adjacency matrix with self-loops

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 $\lambda_1 + \lambda_2 + \lambda_3 + \lambda_4 + \lambda_5 =$ _______.

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Answer: 2 (or 4) (official GATE 2023 final answer key)

Concept used: The sum of the eigenvalues equals the trace; in an adjacency matrix only self-loops sit on the diagonal.

First step: Count the self-loops in the graph: they are the only non-zero diagonal entries of $A$ .

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Source: official GATE 2023 CS question paper and final answer key, organised by IIT Kanpur. Question paper · Answer key