GATE CS 2022 Set 1 — Question 37
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Engineering Mathematics → Graphs (Data Structure) → Adjacency Matrix & List
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Question
Consider a simple undirected unweighted graph with at least three vertices. If is the adjacency matrix of the graph, then the number of 3-cycles in the graph is given by the trace of
Correct answer
(D) A³ divided by 6
Solution
In a simple undirected graph, the number of paths of length from vertex to vertex is given by the -th entry of the matrix . The trace of , denoted as , is the sum of the diagonal elements, which represents the total number of paths of length that start and end at the same vertex.For , a path of length 3 that starts and ends at the same vertex is a 3-cycle (a triangle). In a simple graph, each 3-cycle is counted 6 times in because:
1.It can start at any of the 3 vertices ( or ).
2.For each starting vertex, it can be traversed in 2 directions (e.g., and ).
Therefore, the number of 3-cycles is , which is the trace of .Continue learning with Success Tracker
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