* Hamiltonian path and cycle problems

Hamilton ’ s Icosian Game was a recreational puzzle based on finding a Hamiltonian cycle.

Richard M. Karp showed in 1972 that the Hamiltonian cycle problem was NP-complete, which implies the NP-hardness of TSP.

# Convert the Eulerian circuit of into a Hamiltonian cycle of in the following way: walk along, and each time you are about to come into an already visited vertex, skip it and try to go to the next one ( along ).

The problem of finding a closed knight's tour is similarly an instance of the Hamiltonian cycle problem.

It also forms a Hamiltonian cycle on a hypercube, where each bit is seen as one dimension.

In mathematics, Tait's conjecture states that " Every 3-connected planar cubic graph has a Hamiltonian cycle ( along the edges ) through all its vertices ".

The condition that the graph be 3-regular is necessary due to polyhedra such as the rhombic dodecahedron, which forms a bipartite graph with six degree-four vertices on one side and eight degree-three vertices on the other side ; because any Hamiltonian cycle would have to alternate between the two sides of the bipartition, but they have unequal numbers of vertices, the rhombic dodecahedron is not Hamiltonian.

In a Hamiltonian cubic planar graph, such an edge coloring is easy to find: use two colors alternately on the cycle, and a third color for all remaining edges.

Alternatively, a 4-coloring of the faces of a Hamiltonian cubic planar graph may be constructed directly, using two colors for the faces inside the cycle and two more colors for the faces outside.

If this fragment is part of a larger graph, then any Hamiltonian cycle through the graph must go in or out of the top vertex ( and either one of the lower ones ).

The " compulsory " edges of the fragments, that must be part of any Hamiltonian path through the fragment, are connected at the central vertex ; because any cycle can use only two of these three edges, there can be no Hamiltonian cycle.

* Grinberg's theorem, a necessary condition on the existence of a Hamiltonian cycle that can be used to show that a graph forms a counterexample to Tait's conjecture

In 1982 Vladimir Batagelj and Pisanski proved that the Cartesian product of a tree and a cycle is Hamiltonian if and only if no degree of the tree exceeds the length of the cycle.

A theorem by Nash-Williams says that every < span class =" texhtml " >< var > k </ var ></ span >‑ regular graph on < span class =" texhtml " > 2 < var > k </ var > + 1 </ span > vertices has a Hamiltonian cycle.

In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path or a Hamiltonian cycle exists in a given graph ( whether directed or undirected ).

The questions range from counting ( e. g., the number of graphs on n vertices with k edges ) to structural ( e. g., which graphs contain Hamiltonian cycles ) to algebraic questions ( e. g., given a graph G and two numbers x and y, does the Tutte polynomial T < sub > G </ sub >( x, y ) have a combinatorial interpretation ?).

* Are there any Hamiltonian cycles in a given graph with cost less than 100?

* How many Hamiltonian cycles in a given graph have cost less than 100?

The items may be stored individually as records in a database ; or may be elements of a search space defined by a mathematical formula or procedure, such as the roots of an equation with integer variables ; or a combination of the two, such as the Hamiltonian circuits of a graph.

The knight's tour problem is an instance of the more general Hamiltonian path problem in graph theory.

* Barnette's conjecture, a still-open refinement of Tait's conjecture stating that every bipartite cubic polyhedral graph is Hamiltonian.

Specifically, in quantum mechanics, the state of an atom, i. e. an eigenstate of the atomic Hamiltonian, is approximated by an expansion ( see configuration interaction expansion and basis set ) into linear combinations of anti-symmetrized products ( Slater determinants ) of one-electron functions.

Later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations.

In particular, quantum phase transitions refer to transitions where the temperature is set to zero, and the phases of the system refer to distinct ground states of the Hamiltonian.

In this type of calculation, there is an approximate Hamiltonian and an approximate expression for the total electron density.

Semi-empirical methods follow what are often called empirical methods, where the two-electron part of the Hamiltonian is not explicitly included.

Once the electronic and nuclear variables are separated ( within the Born – Oppenheimer representation ), in the time-dependent approach, the wave packet corresponding to the nuclear degrees of freedom is propagated via the time evolution operator ( physics ) associated to the time-dependent Schrödinger equation ( for the full molecular Hamiltonian ).

In the Hamiltonian formalism, given a coordinate it is possible to derive the appropriate ( generalized ) momentum such that the associated volume is preserved by the flow.

The number of times the interaction Hamiltonian acts is the order of the perturbation expansion, and the time-dependent perturbation theory for fields is known as the Dyson series.

Although the resulting energy eigenfunctions ( the orbitals ) are not necessarily isotropic themselves, their dependence on the angular coordinates follows completely generally from this isotropy of the underlying potential: the eigenstates of the Hamiltonian ( that is, the energy eigenstates ) can be chosen as simultaneous eigenstates of the angular momentum operator.

In quantum mechanics, the Hamiltonian is the operator corresponding to the total energy of the system.

The Hamiltonian is the sum of the kinetic energies of all the particles, plus the potential energy of the particles associated with the system.

For different situations and / or number of particles, the Hamiltonian is different since it includes the sum of kinetic energies of the particles, and the potential energy function corresponding to the situation.

By analogy with classical mechanics, the Hamiltonian is commonly expressed as the sum of operators corresponding to the kinetic and potential energies of a system, in the form

Although this is not the technical definition of the Hamiltonian in classical mechanics, it is the form it most commonly takes.

The general form of the Hamiltonian in this case is:

where the sum is taken over all particles and their corresponding potentials ; the result is that the Hamiltonian of the system is the sum of the separate Hamiltonians for each particle.

( It takes the same form as the Hamilton – Jacobi equation, which is one of the reasons H is also called the Hamiltonian ).

If the Hamiltonian is time-independent,

This follows from rotational invariance of the Hamiltonian, which in turn is guaranteed for a spherically symmetric potential.