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  2. Symmetric graph - Wikipedia

    en.wikipedia.org/wiki/Symmetric_graph

    Additional families of symmetric graphs with an even number of vertices 2n, are the evenly split complete bipartite graphs K n,n and the crown graphs on 2n vertices. Many other symmetric graphs can be classified as circulant graphs (but not all). The Rado graph forms an example of a symmetric graph with infinitely many vertices and infinite degree.

  3. List of graphs - Wikipedia

    en.wikipedia.org/wiki/List_of_graphs

    A gear graph, denoted G n, is a graph obtained by inserting an extra vertex between each pair of adjacent vertices on the perimeter of a wheel graph W n. Thus, G n has 2n+1 vertices and 3n edges. [4] Gear graphs are examples of squaregraphs, and play a key role in the forbidden graph characterization of squaregraphs. [5]

  4. Chromatic symmetric function - Wikipedia

    en.wikipedia.org/wiki/Chromatic_symmetric_function

    The chromatic symmetric function is a symmetric function invariant of graphs studied in algebraic graph theory, a branch of mathematics. It is the weight generating function for proper graph colorings , and was originally introduced by Richard Stanley as a generalization of the chromatic polynomial of a graph.

  5. Symmetry in mathematics - Wikipedia

    en.wikipedia.org/wiki/Symmetry_in_mathematics

    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 onto itself which preserves the structure.

  6. Symmetry group - Wikipedia

    en.wikipedia.org/wiki/Symmetry_group

    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.

  7. Graph automorphism - Wikipedia

    en.wikipedia.org/wiki/Graph_automorphism

    For example, since the graph is symmetric, all edges are equivalent. 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 ]

  8. Automorphisms of the symmetric and alternating groups

    en.wikipedia.org/wiki/Automorphisms_of_the...

    For every symmetric group other than S 6, there is no other conjugacy class consisting of elements of order 2 that has the same number of elements as the class of transpositions. Or as follows: Each permutation of order two (called an involution ) is a product of k > 0 disjoint transpositions, so that it has cyclic structure 2 k 1 n −2 k .

  9. Algebraic graph theory - Wikipedia

    en.wikipedia.org/wiki/Algebraic_graph_theory

    This second branch of algebraic graph theory is related to the first, since the symmetry properties of a graph are reflected in its spectrum. In particular, the spectrum of a highly symmetrical graph, such as the Petersen graph, has few distinct values [ 1 ] (the Petersen graph has 3, which is the minimum possible, given its diameter).