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Cayley graph. In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, [ 1] is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley ), and uses a specified set of generators for the group.
As originally described by Cayley (1846), the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators ( Nikolski 1988 ).
In graph-theoretic terms, the theorem states that for loopless planar graph, its chromatic number is ().. The intuitive statement of the four color theorem – "given any separation of a plane into contiguous regions, the regions can be colored using at most four colors so that no two adjacent regions have the same color" – needs to be interpreted appropriately to be correct.
In group theory, the quaternion group Q 8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset of the quaternions under multiplication. It is given by the group presentation. where e is the identity element and e commutes with the other elements of the group.
Arthur Cayley FRS (/ ˈ k eɪ l i /; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics , and was a professor at Trinity College, Cambridge for 35 years.
Cayley, in his original 1854 paper, [10] showed that the correspondence in the theorem is one-to-one, but he failed to explicitly show it was a homomorphism (and thus an embedding). However, Nummela notes that Cayley made this result known to the mathematical community at the time, thus predating Jordan by 16 years or so.
Then the Cayley graph of with respect to is a locally finite tree and hence a 0-hyperbolic space. Thus F {\displaystyle F} is a hyperbolic group. More generally we see that any group G {\displaystyle G} which acts properly discontinuously on a locally finite tree (in this context this means exactly that the stabilizers in G {\displaystyle G} of ...
Vertex-transitive graph. In the mathematical field of graph theory, a vertex-transitive graph is a graph G in which, given any two vertices v1 and v2 of G, there is some automorphism. such that. In other words, a graph is vertex-transitive if its automorphism group acts transitively on its vertices. [ 1] A graph is vertex-transitive if and only ...