**Graph Trails and Paths****Q6C. Consider the graph with the following vertices and edges:**- **Vertices (V):** {a, b, c, d, e, f}- **Edges (E):** {{a, b}, {a, c}, {a, d}, {a, f}, {b, e}, {b, f}, {c, d}, {d, e}, {d, f}}The graph is displayed with six vertices labeled a, b, c, d, e, and f. Lines connecting the vertices represent the edges.**Visual Description of the Graph:**- Vertex a is connected to vertices b, c, d, and f.- Vertex b is connected to vertices a, e, and f.- Vertex c is connected to vertices a and d.- Vertex d is connected to vertices a, c, e, and f.- Vertex e is connected to vertices b and d.- Vertex f is connected to vertices a, b, and d.**Which of the following are examples of trails within the graph? (Select all that apply.)**- [ ] < a, b, f, d, c, a >- [ ] < d, c, a, b, e >- [ ] < a, b, c, d, e >- [ ] < b, f, a, b >- [ ] < f, b, e >A trail in a graph is a walk that does not repeat any edges. Analyze each sequence to determine whether it represents a valid trail. **Q5C. Consider the graph with the following vertices and edges:**- **Vertices (V):** {a, b, c, d, e, f, g, h, i}- **Edges (E):** {{a, b}, {a, c}, {a, d}, {a, i}, {b, c}, {b, e}, {b, i}, {c, d}, {d, f}, {d, h}, {e, f}, {e, i}, {f, g}, {g, h}, {h, i}}**Task:**Explain why the graph either does or does not have an Euler trail.**Graph Explanation:**The accompanying diagram represents a graph with vertices labeled from 'a' to 'i,' connected by various edges. The connections between specified pairs form a complex network. To determine if there is an Euler trail, the degrees (number of connections) of each vertex must be considered.**Multiple Choice Options:**- ( ) All vertices have even degree. Therefore, the graph does not have an Euler trail.- ( ) Exactly two vertices (b and e) have odd degree. Therefore, the graph does have an Euler trail.- ( ) Exactly two vertices (c and e) have odd degree. Therefore, the graph does have an Euler trail.- ( ) Four vertices (a, b, d, and e) have odd degree. Therefore, the graph does not have an Euler trail.- ( ) All vertices have odd degree. Therefore, the graph does not have an Euler trail.

Since you have asked multiple questions, we will solve the first question for you. If you want any specific question to be solved then please specify the question number or post only that question. Q6C A walk is a sequence of vertices and edges of a graph. A trail is a walk in which no edge is repeated. In the first choice a,b,f,d,c,a, the edges are a,b, b,f,f,d, d,c,c,a. All the edges are in E. None of the edges repeat. Hence, a,b,f,d,c,a is a trail. In the second choice d,c,a,b,e, the edges are d,c,c,a,a,b,b,e. None of the edges repeat. Hence, d,c,a,b,e is a trail.In the third choice a,b,c,d,e, the edges are a,b, b,c,c,d, d,e. The edge b,c is not in E. Hence, a,b,c,d,e is not a trail. In the fourth choice b,f,a,b, the edges are b,f, f,a,a,b. All the edges are in E. None of the edges repeat. Hence, b,f,a,b is a trail. In the fifth choice f,b,e , the edges are f,b,b,e. All the edges are in E. None of the edges repeat. Hence, f,b,e is a trail. Therefore, all choices other than a,b,c,d,e are trails.