The problem asks you to draw a directed graph (or digraph) $E$ based on the given sets:

1. Vertices (V):
   $V = \{v_1, v_2, v_3, v_4, v_5\}$

2. Edges (E):
   $E = \{(v_1, v_2), (v_2, v_3), (v_3, v_4), (v_4, v_1), (v_5, v_5), (v_5, v_1)\}$

Steps to draw the directed graph:

1. Vertices: Place 5 vertices labeled $v_1, v_2, v_3, v_4, v_5$.
2. Edges: Connect the vertices with directed edges (arrows) based on the following pairs:
   • $(v_1, v_2)$: Draw an arrow from $v_1$ to $v_2$.
   • $(v_2, v_3)$: Draw an arrow from $v_2$ to $v_3$.
   • $(v_3, v_4)$: Draw an arrow from $v_3$ to $v_4$.
   • $(v_4, v_1)$: Draw an arrow from $v_4$ to $v_1$, forming a cycle with $v_1, v_2, v_3, v_4$.
   • $(v_5, v_5)$: Draw a loop from $v_5$ to itself.
   • $(v_5, v_1)$: Draw an arrow from $v_5$ to $v_1$.

This will result in a directed graph where:

• There is a cycle connecting $v_1, v_2, v_3, v_4$,
• $v_5$ has a self-loop and is connected to $v_1$.

Would you like a detailed image of the graph or further clarification?

Here are 5 related questions to expand your understanding:

1. What is the difference between directed and undirected graphs?
2. How do you identify cycles in directed graphs?
3. What are the in-degrees and out-degrees of each vertex in this graph?
4. What are the practical applications of directed graphs?
5. How do self-loops affect the structure of a graph?

Tip: In a directed graph, the direction of edges is crucial for understanding relationships between vertices, unlike in an undirected graph.