The adjacency matrix of a bipartite graph can be partitioned into:A. Two square submatricesB. Two rectangular submatricesC. One square and one rectangular submatrixD. Two identical submatrices
Question
The adjacency matrix of a bipartite graph can be partitioned into:A. Two square submatricesB. Two rectangular submatricesC. One square and one rectangular submatrixD. Two identical submatrices
Solution
The correct answer is B. Two rectangular submatrices.
In a bipartite graph, the vertices can be divided into two disjoint and independent sets, U and V. Each edge connects a vertex in U to one in V. This means that there are no edges within the sets U or V.
When we represent this graph as an adjacency matrix, we can partition the matrix into four submatrices. The submatrices on the diagonal represent the connections within the sets U and V. Since there are no such connections in a bipartite graph, these submatrices are zero matrices.
The off-diagonal submatrices represent the connections between the sets U and V. These are rectangular matrices, because the number of vertices in U and V may not be the same.
So, the adjacency matrix of a bipartite graph can be partitioned into two rectangular submatrices.
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