Hamiltonian cycle polynomial

The Hamiltonian cycle polynomial, often referred to in the context of graph theory, is a polynomial associated with a graph that encodes information about the Hamiltonian cycles of that graph. A Hamiltonian cycle is a cycle that visits every vertex in the graph exactly once and returns to the starting vertex. To define the Hamiltonian cycle polynomial for a graph \(G\), we denote it as \(H(G, x, y)\).