The image contains a directed graph (digraph) with six vertices labeled $1$ through $6$, along with several questions about the properties of the graph. Here's the breakdown of each question:

(a) Quel est son ordre ? Justifier.

The order of a graph is the number of vertices. To determine the order of this graph, count the vertices.

Answer: The graph has 6 vertices, so its order is 6.

(b) Est-il régulier ? Justifier.

A graph is regular if every vertex has the same number of edges (degree). For a directed graph, there are both incoming edges (in-degree) and outgoing edges (out-degree).

Answer: To determine if the graph is regular, we need to check if all vertices have the same in-degree and out-degree.

• Vertex $1$ has an out-degree of 2 (edges to 2 and 5) and in-degree of 2 (edges from 2 and 6).
• Vertex $2$ has an out-degree of 2 (edges to 3 and 6) and in-degree of 2 (edges from 1 and 5).
• Vertex $3$ has an out-degree of 1 (edge to 4) and in-degree of 2 (edges from 2 and 4).
• Vertex $4$ has an out-degree of 2 (edges to 3 and 5) and in-degree of 2 (edges from 3 and 6).
• Vertex $5$ has an out-degree of 1 (edge to 6) and in-degree of 2 (edges from 1 and 4).
• Vertex $6$ has an out-degree of 1 (edge to 1) and in-degree of 2 (edges from 2 and 5).

Since the degrees are not the same for all vertices (e.g., vertex 3 has an out-degree of 1 while others have 2), the graph is not regular.

(c) Est-il connexe ? Justifier.

A directed graph is strongly connected if there is a directed path between any pair of vertices. We need to verify if all vertices can be reached from any other vertex following the direction of the edges.

Answer: Looking at the graph, it is possible to trace paths between certain vertices, but some vertices do not have directed paths connecting them both ways (e.g., from $6$ to $3$, no direct path exists). Therefore, the graph is not strongly connected.

(d) Quel est le plus court chemin (PCC) entre $1$ et $5$ ? Justifier.

The shortest path between vertices $1$ and $5$ is the path with the fewest edges. To find it, we trace possible routes.

Answer: The shortest path from $1$ to $5$ is through vertex $4$: $1 \to 2 \to 6 \to 4 \to 5$. This path has 4 edges.

(e) Quelle est la longueur et la capacité de ce plus court chemin ? Justifier.

The length of a path in a graph is the number of edges in the path, and capacity refers to additional weights or values associated with the edges (not specified in the image, so we assume a simple graph without edge capacities).

Answer: The length of the shortest path from $1$ to $5$ is 4. Since there are no capacities specified, the capacity is undefined or irrelevant in this case.

Let me know if you need more details or explanations!

5 Related Questions:

1. What is the definition of an Eulerian path in a directed graph?
2. How would the graph change if an additional edge were added between vertices 3 and 6?
3. What conditions must be satisfied for a directed graph to be regular?
4. How can you verify if a directed graph is strongly connected using matrix representation?
5. What is Dijkstra's algorithm, and how can it be applied to find the shortest path in a weighted directed graph?

Tip:

When analyzing directed graphs, it's useful to differentiate between in-degrees and out-degrees for a clearer understanding of connectivity and regularity.