To analyze whether each graph has a Hamiltonian circuit, let's recall that a Hamiltonian circuit is a cycle that visits each vertex of a graph exactly once and returns to the starting vertex.

Let's examine each graph from left to right:

1. First Graph (Square with Diagonal Edge):

   • This graph has four vertices arranged in a square with an additional diagonal connecting two opposite vertices.
   • A Hamiltonian circuit exists here because you can start at any vertex, move around the square visiting each vertex exactly once, and return to the starting point. One possible Hamiltonian circuit is: $A \rightarrow B \rightarrow D \rightarrow C \rightarrow A$.

2. Second Graph (Pentagon with Central Vertex):

   • This graph has a central vertex connected to each of the five outer vertices that form a pentagon.
   • A Hamiltonian circuit does not exist in this graph. The central vertex would need to be visited exactly once within a cycle, but because it is connected to every other vertex, a path including it would prevent a circuit that visits each outer vertex exactly once. Thus, no Hamiltonian circuit exists.

3. Third Graph (Bowtie or Hourglass Shape):

   • This graph consists of two triangles sharing a common central vertex.
   • A Hamiltonian circuit does not exist in this graph. Due to the shared vertex between the two triangles, any cycle that attempts to visit all vertices exactly once would either skip vertices or revisit some. Therefore, a Hamiltonian circuit is not possible.

Summary:

• First Graph: Has a Hamiltonian circuit.
• Second Graph: No Hamiltonian circuit.
• Third Graph: No Hamiltonian circuit.

Would you like more details on any specific part or need help with similar questions?

Further Questions:

1. What is the difference between a Hamiltonian circuit and an Eulerian circuit?
2. How can we determine if a graph is Hamiltonian by looking at its vertices and edges?
3. Are there any specific algorithms to find Hamiltonian circuits in complex graphs?
4. Can a graph with a Hamiltonian path lack a Hamiltonian circuit? Why or why not?
5. How does graph connectivity affect the existence of a Hamiltonian circuit?

Tip:

Remember, for a Hamiltonian circuit to exist, each vertex must be visited exactly once in a single cycle, returning to the start. Often, if a graph has a vertex of degree 1 (only one edge), it cannot be Hamiltonian.