GATE CS 2024 Set 2 — Question 17
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Engineering Mathematics → Graphs (Data Structure) → Adjacency Matrix & List
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Question
Let be the adjacency matrix of a simple undirected graph . Suppose is its own inverse. Which one of the following statements is always TRUE?
Correct answer
(B) G is a perfect matching
Solution
Let be the adjacency matrix of a simple undirected graph . Since is simple, the diagonal entries of are all 0 (). Since is undirected, is symmetric ().Given that is its own inverse, we have , which implies (the identity matrix).The diagonal entries of represent the degree of each vertex in the graph. Specifically:Since , . Thus, is the sum of the entries in row , which is the degree of vertex .Since , we must have for all . Therefore, every vertex in has a degree of exactly 1.A graph where every vertex has a degree of 1 is a collection of disjoint edges covering all vertices, which corresponds to a perfect matching.
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