In this note we show how the Hamiltonian Cycle problem can be reduced to solving a system of polynomial equations related to the adjacency matrix of a graph. This system of equations can be solved using the method of Gröbner bases, but we also show how a symbolic determinant related to the adjacency matrix can be used to directly decide whether a graph has a Hamiltonian cycle.
"SOLVING THE HAMILTONIAN CYCLE PROBLEM USING SYMBOLIC DETERMINANTS." Taiwanese J. Math. 10 (2) 327 - 338, 2006. https://doi.org/10.11650/twjm/1500403828