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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.
This is usually done by studying the Cayley graphs of groups, which, in addition to the graph structure, are endowed with the structure of a metric space, given by the so-called word metric. Geometric group theory, as a distinct area, is relatively new, and became a clearly identifiable branch of mathematics in the late 1980s and early 1990s.
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}} .
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 ...
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 ...
A Cayley graph of the symmetric group S 4 using the generators (red) a right circular shift of all four set elements, and (blue) a left circular shift of the first three set elements. Cayley table, with header omitted, of the symmetric group S 3. The elements are represented as matrices. To the left of the matrices, are their two-line form.
A college student just solved a seemingly paradoxical math problem—and the answer came from an incredibly unlikely place. Skip to main content. 24/7 Help. For premium support please call: 800 ...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.