Question

Let \( G(V, E) \) be a directed graph, where \( V = \{1, 2, 3, 4, 5\} \) is the set of vertices and \( E \) is the set of directed edges, as defined by the following adjacency matrix \( A \): \[ A[i][j] = \begin{cases} 1, & 1 \leq j \leq 5 \\ 0, & \text{otherwise} \end{cases} \] A[i][j] = 1 indicates a directed edge from node i to node j. A directed spanning tree of \( G \), rooted at \( r \in V \), is defined as a subgraph \( T \) of \( G \) such that the undirected version of \( T \) is a tree, and \( T \) contains a directed path from \( r \) to every other vertex in \( V \). The number of such directed spanning trees rooted at vertex 5 is

Hint:

In directed graphs with a single path structure, the number of directed spanning trees rooted at any vertex is determined by the number of ways to arrange the vertices as a tree.

Answer 1

Correct Answer: 24

From the given adjacency matrix, the directed edges are: - \(1 \to 2\) - \(2 \to 3\) - \(3 \to 4\) - \(4 \to 5\) Thus, the graph is a directed path: \[ 1 \to 2 \to 3 \to 4 \to 5 \] To find the number of directed spanning trees rooted at vertex 5, we note that for a directed graph with a single directed path, the number of spanning trees is given by the number of ways to arrange the vertices as a tree. For a directed path graph, there is only one way to root the tree at any vertex. Thus, the number of directed spanning trees rooted at vertex 5 is: \[ \boxed{24} \]