Search results
Results from the WOW.Com Content Network
The smallest connected half-transitive graph is Holt's graph, with degree 4 and 27 vertices. [1] [6] Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex-transitive and edge-transitive, rather than an arc-transitive graph. Such a definition would include half-transitive graphs, which are excluded under the ...
A skew-symmetric graph may equivalently be defined as the double covering graph of a polar graph or switch graph, [1] which is an undirected graph in which the edges incident to each vertex are partitioned into two subsets. Each vertex of the polar graph corresponds to two vertices of the skew-symmetric graph, and each edge of the polar graph ...
The easier problem of testing whether a graph has any symmetries (nontrivial automorphisms), known as the graph automorphism problem, also has no known polynomial time solution. [5] There is a polynomial time algorithm for solving the graph automorphism problem for graphs where vertex degrees are bounded by a constant. [ 6 ]
Another example of a symmetry group is that of a combinatorial graph: a graph symmetry is a permutation of the vertices which takes edges to edges. Any finitely presented group is the symmetry group of its Cayley graph; the free group is the symmetry group of an infinite tree graph.
The symmetry factor theorem gives the symmetry factor for a general diagram: the contribution of each Feynman diagram must be divided by the order of its group of automorphisms, the number of symmetries that it has. An automorphism of a Feynman graph is a permutation M of the lines and a permutation N of the vertices with the following properties:
Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after rotation of 180 degrees about the origin. If = is in the domain of an odd function (), then () =. Examples of odd functions are:
The points within an orbit are "equivalent". If a symmetry group applies for a pattern, then within each orbit the color is the same. The set of all orbits of X under the action of G is written as X / G. If Y is a subset of X, we write GY for the set { g · y : y ∈ Y and g ∈ G}. We call the subset Y invariant under G if GY = Y (which is ...
Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations. [1] Given a structured object X of any sort, a symmetry is a mapping of the object