Search results
Results from the WOW.Com Content Network
The Hirotaka graph, discovered by Hirotaka Yoneda, consists of 7 nodes and 12 edges, and is the minimal and unique Harris graph. The Hirotaka graph, with 7 and size 12, is the Harris graph with the smallest order. [1] [2] Douglas Shaw proved it to be minimal by showing all Eulerian graphs of order 6 or lower were not Hamiltonian and tough.
A connected graph has an Euler cycle if and only if every vertex has an even number of incident edges. The term Eulerian graph has two common meanings in graph theory. One meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of even degree. These definitions coincide for connected graphs. [2]
In the Eulerian specification of a field, the field is represented as a function of position x and time t. For example, the flow velocity is represented by a function (,). On the other hand, in the Lagrangian specification, individual fluid parcels are followed through time.
In one direction, the Hamiltonian path problem for graph G can be related to the Hamiltonian cycle problem in a graph H obtained from G by adding a new universal vertex x, connecting x to all vertices of G. Thus, finding a Hamiltonian path cannot be significantly slower (in the worst case, as a function of the number of vertices) than finding a ...
The Petersen graph is hypo-Hamiltonian: by deleting any vertex, such as the center vertex in the drawing, the remaining graph is Hamiltonian. This drawing with order-3 symmetry is the one given by Kempe (1886). The Petersen graph has a Hamiltonian path but no Hamiltonian cycle. It is the smallest bridgeless cubic graph with no Hamiltonian cycle.
A connected planar graph is Eulerian (has even degree at every vertex) if and only if its dual graph is bipartite. [32] A Hamiltonian cycle in a planar graph G corresponds to a partition of the vertices of the dual graph into two subsets (the interior and exterior of the cycle) whose induced subgraphs are both trees.
Each n-dimensional De Bruijn graph is the line digraph of the (n – 1)-dimensional De Bruijn graph with the same set of symbols. [4] Each De Bruijn graph is Eulerian and Hamiltonian. The Euler cycles and Hamiltonian cycles of these graphs (equivalent to each other via the line graph construction) are De Bruijn sequences.
However, the semi-implicit Euler method is a symplectic integrator, unlike the standard method. As a consequence, the semi-implicit Euler method almost conserves the energy (when the Hamiltonian is time-independent). Often, the energy increases steadily when the standard Euler method is applied, making it far less accurate.