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Let's consider a simple graph with 5 vertices and form a shape similar to a pentagon with an extra edge connecting two non-adjacent vertices. Illustrating this graph as follows: 1 --- 2 | / \ 5 -- 4 -- 3 We can represent this graph with this adjacency list: 1: 2, 5 2: 1, 3, 4 3: 2, 4 4: 2, 3, 5 5: 1, 4 Now we will analyze this graph to check whether it is Eulerian or Hamiltonian.

To check if the graph is Eulerian, we need to ensure that all vertices have even degree. From our adjacency list, we can see the degree of each vertex: 1: degree 2 2: degree 3 3: degree 2 4: degree 3 5: degree 2 We can observe that vertices 2 and 4 have an odd degree, so the graph is not Eulerian.

For a graph to be Hamiltonian, there must be a Hamiltonian cycle. However, finding a Hamiltonian cycle can be difficult, and there is no simple condition to check like the Eulerian graphs. In this case, we can manually search for a Hamiltonian cycle by looking for a path that visits every vertex exactly once and returns to the starting vertex. No matter how we visit the vertices, we will not be able to find a Hamiltonian cycle because it will not be possible to visit vertex 5 without visiting either vertex 1 or 4 twice. For example, if we start at vertex 1, we can only move to vertex 2 next and then to vertices 3 and 4 in any order. After visiting vertices 3 and 4, we will be forced to visit either vertex 1 or 4 again, violating the Hamiltonian condition. Therefore, this graph is neither Eulerian nor Hamiltonian.