https://file.aiquickdraw.com/mathbot/file/1727278081851qnuwiazh.jpg

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.

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### 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.

We are given a directed graph (digraph) with six vertices labeled from 1 to 6, and asked several questions about its properties, including order, regularity, connectedness, shortest path, and its length and capacity.

Analyze a directed graph with six vertices and solve questions related to its order, regularity, connectivity, and the shortest path between two vertices. This problem covers important graph theory concepts suitable for undergraduate students.

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