Discussion
Let \(A\) be the adjacency matrix of a simple undirected graph \(G\). Suppose \(A\) is its own inverse. Which one of the following statements is always TRUE?
Answer: B
The statement that is always true is \(G\) is a perfect matching.
A symmetric matrix that is its own inverse satisfies \(A^{2} = I\). For an adjacency matrix, the diagonal entry \((A^{2})_{ii}\) is the degree of vertex i, so every vertex has degree exactly 1. An off-diagonal entry \((A^{2})_{ij}\) counts paths of length 2 from i to j, and these are 0 automatically when every degree is 1. A graph in which every vertex has exactly one edge is a perfect matching.
A cycle has every degree 2, and a complete graph on more than two vertices has degree \(n - 1 \gt 1\), so neither works (except the single edge \(K_2\), which is itself a matching). Since a perfect matching does satisfy \(A^{2} = I\), such graphs exist.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development