The image contains a set of graph-related problems for a university-level Graph Theory exam. Here's a breakdown of the questions and instructions to solve them:

Questions:

1. Explain the history of graph theory and provide real-life applications of graph theory in daily life.

2. Construct the adjacency matrix for the given graph.
   (Graph is shown with nodes A, B, C, D, E and directed edges.)

3. Find the shortest path from node A to node E using the graph provided:
   (Includes weighted edges between nodes A, B, C, D, E.)

   • a. Determine the total number of possible paths from A to E.
   • b. Calculate the total weights for each path from A to E.
   • c. Identify the shortest path from A to E and its weight.

4. Determine the Eulerian path of the given graph.
   (Another graph with nodes 1 through 6 and edges is provided.)

5. Using Dijkstra's Algorithm, find the shortest distance and path from node A to node E.
   (Graph with nodes A, B, C, D, E and weights is provided.)

Solutions Approach:

1. History of Graph Theory and Applications:

   • Graph theory began with Leonhard Euler in 1736 with the famous Königsberg Bridge Problem.
   • Applications: Social networks, transportation networks, circuit designs, and more.

2. Adjacency Matrix:

   • Create a matrix where rows and columns represent nodes, and entries represent the edge weight (or 0 if no edge).

3. Shortest Path (Paths from A to E):

   • Enumerate all paths from A to E using the graph.
   • Sum the weights for each path.
   • Use algorithms like Breadth-First Search (BFS) or manual checks to determine the shortest path.

4. Eulerian Path:

   • Check conditions: An Eulerian path exists if at most two vertices have odd degrees.
   • Trace the Eulerian path manually.

5. Dijkstra's Algorithm:

   • Use the algorithm step-by-step:
     1. Assign initial distances (0 for the starting node, infinity for others).
     2. Update distances as you explore neighbors.
     3. Finalize shortest distances and trace the shortest path.

Would you like me to solve one of these problems step-by-step, or explain a particular concept in detail? Let me know how you'd like to proceed! Here are 5 related follow-up questions:

Related Questions:

1. What are the key differences between adjacency matrices and adjacency lists in graph representation?
2. How do you identify if a graph is Eulerian or Semi-Eulerian?
3. Can Dijkstra's Algorithm handle graphs with negative edge weights? Why or why not?
4. What are some other algorithms for finding the shortest path in a graph besides Dijkstra's?
5. How can graph theory be applied to model real-world scheduling problems?

Tip:

When working with adjacency matrices, remember that directed graphs are asymmetric, meaning the matrix is not necessarily symmetric.