The adjacency matrix of directed acyclic powerless graphs at n vertices

The following answers are quoted from GPT-3 model, please use them wisely:

示例：

答案：B n(n-1)2

这是一个有向无环无权图，其邻接矩阵中元素只有0和1，所以邻接矩阵中至少有 n(n-1)2 个零元素。

例如，一个有5个顶点的有向无环无权图的邻接矩阵表示如下：

      1  2  3  4  5
  1   0  1  0  0  1
  2   0  0  1  0  0
  3   0  0  0  1  0
  4   0  0  0  0  1
  5   0  0  0  0  0

可以看出，邻接矩阵中共有 12 个零元素，即为 n(n-1)2，符合答案B n(n-1)2。

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B

In a directed acyclic unbounded graph, there is no loop from one vertex to that vertex through several edges, so in the adjacency matrix of the graph, the element on the diagonal must be 0.

Because the graph is undirected, for every element a[i][j] in the matrix, if a[i][j] is 1, then a[j][i] must be 0(otherwise there is a loop from j to i), so one of the elements of the upper and lower triangles of the matrix must be 0.

Therefore, the adjacency matrix of the directed acyclic powerless graph has at least frac{n(n-1)}{2} zero elements.

Therefore, choice B n(n-1)/2 is correct.