Presentation Type
Poster
Presentation Type
Submission
Keywords
Hamiltonian cycles, circulant graphs, counting, digraph, winding number, generating function, edges, and circuit
Department
Mathematics
Major
Mathematics
Abstract
We consider the problem of counting Hamiltonian cycles in circulant graphs $C^K_n$ where $n$ is the number of vertices and $K$ is a set containing elements that correspond to the allowed edges in the circulant graphs. After sorting the cycles by a topological invariant called the winding number, we use a modified transfer matrix method to convert local data into global structures. The result is a generating function that counts the number of Hamiltonian cycles in a circulant graph with $n$ vertices. The results for $K=\{1,2\}$ and $K=\{1,3\}$ have been found by previous authors. We focus on the case where $K=\{1,4\}$, but we continue to extend our research to generalize for all $K$.
Faculty Mentor
Joshua Bowman
Funding Source or Research Program
Academic Year Undergraduate Research Initiative, Summer Undergraduate Research Program, Undergraduate Research Fellowship
Location
Waves Cafeteria
Start Date
10-4-2026 1:00 PM
End Date
10-4-2026 2:00 PM
Counting Hamiltonian Cycles in Quartic Circulant Graphs
Waves Cafeteria
We consider the problem of counting Hamiltonian cycles in circulant graphs $C^K_n$ where $n$ is the number of vertices and $K$ is a set containing elements that correspond to the allowed edges in the circulant graphs. After sorting the cycles by a topological invariant called the winding number, we use a modified transfer matrix method to convert local data into global structures. The result is a generating function that counts the number of Hamiltonian cycles in a circulant graph with $n$ vertices. The results for $K=\{1,2\}$ and $K=\{1,3\}$ have been found by previous authors. We focus on the case where $K=\{1,4\}$, but we continue to extend our research to generalize for all $K$.