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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.
Visualization comparing the sheet and the binary tree Cayley graph of (,). Red and blue edges correspond to a {\displaystyle a} and b {\displaystyle b} , respectively. In the mathematical field of group theory , the Baumslag–Solitar groups are examples of two-generator one-relator groups that play an important role in combinatorial group ...
Geometric group theory grew out of combinatorial group theory that largely studied properties of discrete groups via analyzing group presentations, which describe groups as quotients of free groups; this field was first systematically studied by Walther von Dyck, student of Felix Klein, in the early 1880s, [2] while an early form is found in the 1856 icosian calculus of William Rowan Hamilton ...
Hence, the fundamental group of the Cayley graph Γ(G) is isomorphic to the kernel of φ, the normal subgroup of relations among the generators of G. The extreme case is when G = {e}, the trivial group, considered with as many generators as F, all of them trivial; the Cayley graph Γ(G) is a bouquet of circles, and its fundamental group is F ...
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.
Download QR code; Print/export Download as PDF; Printable version; In other projects ... Pages in category "Cayley graphs" The following 3 pages are in this category ...
In mathematics, Cayley's formula is a result in graph theory named after Arthur Cayley. It states that for every positive integer n {\displaystyle n} , the number of trees on n {\displaystyle n} labeled vertices is n n − 2 {\displaystyle n^{n-2}} .
The Cayley graph of a free group with two generators. This is a hyperbolic group whose Gromov boundary is a Cantor set. Hyperbolic groups and their boundaries are important topics in geometric group theory, as are Cayley graphs. The (6,4,2) triangular hyperbolic tiling. The triangle group corresponding to this tiling has a circle as its Gromov ...