Concept
Cayley graph
Related concepts (36)
Max Dehn
Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Dehn's early life and career took place in Germany. However, he was forced to retire in 1935 and eventually fled Germany in 1939 and emigrated to the United States. Dehn was a student of David Hilbert, and in his habilitation in 1900 Dehn resolved Hilbert's third problem, making him the first to resolve one of Hilbert's well-known 23 problems.
Regular graph
In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each internal vertex are equal to each other. A regular graph with vertices of degree k is called a k‐regular graph or regular graph of degree k. Also, from the handshaking lemma, a regular graph contains an even number of vertices with odd degree.
Conjugacy class
In mathematics, especially group theory, two elements and of a group are conjugate if there is an element in the group such that This is an equivalence relation whose equivalence classes are called conjugacy classes. In other words, each conjugacy class is closed under for all elements in the group. Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure.
Combinatorial group theory
In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations. It is much used in geometric topology, the fundamental group of a simplicial complex having in a natural and geometric way such a presentation. A very closely related topic is geometric group theory, which today largely subsumes combinatorial group theory, using techniques from outside combinatorics besides.
Word metric
In group theory, a word metric on a discrete group is a way to measure distance between any two elements of . As the name suggests, the word metric is a metric on , assigning to any two elements , of a distance that measures how efficiently their difference can be expressed as a word whose letters come from a generating set for the group. The word metric on G is very closely related to the Cayley graph of G: the word metric measures the length of the shortest path in the Cayley graph between two elements of G.
Cycle graph (algebra)
In group theory, a subfield of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. A cycle is the set of powers of a given group element a, where an, the n-th power of an element a is defined as the product of a multiplied by itself n times. The element a is said to generate the cycle. In a finite group, some non-zero power of a must be the group identity, e; the lowest such power is the order of the cycle, the number of distinct elements in it.
Spectral graph theory
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. The adjacency matrix of a simple undirected graph is a real symmetric matrix and is therefore orthogonally diagonalizable; its eigenvalues are real algebraic integers. While the adjacency matrix depends on the vertex labeling, its spectrum is a graph invariant, although not a complete one.
Expander graph
In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander constructions have spawned research in pure and applied mathematics, with several applications to complexity theory, design of robust computer networks, and the theory of error-correcting codes. Intuitively, an expander graph is a finite, undirected multigraph in which every subset of the vertices that is not "too large" has a "large" boundary.
Cartesian product of graphs
In graph theory, the Cartesian product G □ H of graphs G and H is a graph such that: the vertex set of G □ H is the Cartesian product V(G) × V(H); and two vertices (u,u' ) and (v,v' ) are adjacent in G □ H if and only if either u = v and u' is adjacent to v' in H, or u' = v' and u is adjacent to v in G. The Cartesian product of graphs is sometimes called the box product of graphs [Harary 1969]. The operation is associative, as the graphs (F □ G) □ H and F □ (G □ H) are naturally isomorphic.
Arthur Cayley
Arthur Cayley (ˈkeɪli; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, Cayley enjoyed solving complex maths problems for amusement. He entered Trinity College, Cambridge, where he excelled in Greek, French, German, and Italian, as well as mathematics. He worked as a lawyer for 14 years. He postulated what is now known as the Cayley–Hamilton theorem—that every square matrix is a root of its own characteristic polynomial, and verified it for matrices of order 2 and 3.