GATE CS 2023 Set 1 — Question 30
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Engineering Mathematics → Graphs (Data Structure) → Adjacency Matrix & List
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Question
Let be the adjacency matrix of the graph with vertices {1, 2, 3, 4, 5}.
Let , and be the five eigenvalues of . Note that these eigenvalues need not be distinct.The value of .
Correct answer
2 to 2
Solution
The sum of the eigenvalues of a square matrix is equal to the trace of the matrix, which is the sum of its diagonal elements:In an adjacency matrix of a graph, the diagonal element represents the number of self-loops at vertex . Looking at the provided graph:
- Vertex 1 has no self-loop ().
- Vertex 2 has no self-loop ().
- Vertex 3 has one self-loop ().
- Vertex 4 has one self-loop ().
- Vertex 5 has no self-loop ().
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