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# Hypercube graph

In graph theory, the hypercube graph Qn is a regular graph with 2n vertices, 2n&minus;1n edges, and n edges touching each vertex. [1]

61 relations: Balinski's theorem, Bijection, Binary number, Bipartite graph, Boolean algebra (structure), Carla Savage, Cartesian product of graphs, Complete coloring, Complete graph, Cube, Cube-connected cycles, Cubic graph, Cycle (graph theory), De Bruijn graph, Disjoint union, Distance-regular graph, Edge (geometry), Expander graph, Fibonacci cube, Folded cube graph, Genus (mathematics), Glossary of graph theory, Graph bandwidth, Graph coloring, Graph drawing, Graph theory, Gray code, Halved cube graph, Hamiltonian path, Hamming distance, Hamming graph, Hasse diagram, Hypercube, Journal of Combinatorial Theory, K-vertex-connected graph, Lévy family of graphs, Levi graph, Longest path problem, Matching (graph theory), Mathematical induction, Möbius configuration, Median graph, N-skeleton, Network topology, Pancyclic graph, Partial cube, Path (graph theory), Permutation, Planar graph, Regular graph, ... Expand index (11 more) »

## Balinski's theorem

In polyhedral combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of three-dimensional polyhedra and higher-dimensional polytopes.

## Bijection

In mathematics, a bijection, bijective function or one-to-one correspondence is a function between the elements of two sets, where every element of one set is paired with exactly one element of the other set, and every element of the other set is paired with exactly one element of the first set.

## Binary number

In mathematics and digital electronics, a binary number is a number expressed in the binary numeral system, or base-2 numeral system, which represents numeric values using two different symbols: typically 0 (zero) and 1 (one).

## Bipartite graph

In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint sets U and V (that is, U and V are each independent sets) such that every edge connects a vertex in U to one in V. Vertex set U and V are often denoted as partite sets.

## Boolean algebra (structure)

In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.

## Carla Savage

Carla Diane Savage is an American computer scientist and mathematician, a professor of computer science at North Carolina State University and the secretary of the American Mathematical Society.

## Cartesian product of graphs

In graph theory, the Cartesian product G \square H of graphs G and H is a graph such that.

## Complete coloring

In graph theory, complete coloring is the opposite of harmonious coloring in the sense that it is a vertex coloring in which every pair of colors appears on at least one pair of adjacent vertices.

No description.

## Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

## Cube-connected cycles

In graph theory, the cube-connected cycles is an undirected cubic graph, formed by replacing each vertex of a hypercube graph by a cycle.

## Cubic graph

In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three.

## Cycle (graph theory)

In graph theory, there are several different types of object called cycles, principally a closed walk and a simple cycle; also, e.g., an element of the cycle space of the graph.

## De Bruijn graph

In graph theory, an n-dimensional De Bruijn graph of m symbols is a directed graph representing overlaps between sequences of symbols.

## Disjoint union

In set theory, the disjoint union (or discriminated union) of a family of sets is a modified union operation that indexes the elements according to which set they originated in.

## Distance-regular graph

In mathematics, a distance-regular graph is a regular graph such that for any two vertices v and w, the number of vertices at distance j from v and at distance k from w depends only upon j, k, and i.

## Edge (geometry)

In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope.

## Expander graph

In combinatorics, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion as described below.

## Fibonacci cube

The Fibonacci cubes or Fibonacci networks are a family of undirected graphs with rich recursive properties derived from its origin in number theory.

## Folded cube graph

In graph theory, a folded cube graph is an undirected graph formed from a hypercube graph by adding to it a perfect matching that connects opposite pairs of hypercube vertices.

## Genus (mathematics)

In mathematics, genus (plural genera) has a few different, but closely related, meanings.

## Glossary of graph theory

Graph theory is a growing area in mathematical research, and has a large specialized vocabulary.

## Graph bandwidth

In graph theory, the graph bandwidth problem is to label the n vertices vi of a graph G with distinct integers f(vi) so that the quantity \max\ is minimized (E is the edge set of G).

## Graph coloring

In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints.

## Graph drawing

Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graphs arising from applications such as social network analysis, cartography, and bioinformatics.

## Graph theory

In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.

## Gray code

The reflected binary code, also known as Gray code after Frank Gray, is a binary numeral system where two successive values differ in only one bit (binary digit).

## Halved cube graph

In graph theory, the halved cube graph or half cube graph of order n is the graph of the demihypercube, formed by connecting pairs of vertices at distance exactly two from each other in the hypercube graph.

## Hamiltonian path

In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once.

## Hamming distance

In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different.

## Hamming graph

Hamming graphs are a special class of graphs named after Richard Hamming and used in several branches of mathematics and computer science.

## Hasse diagram

In order theory, a Hasse diagram (German) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction.

## Hypercube

In geometry, a hypercube is an n-dimensional analogue of a square (n.

## Journal of Combinatorial Theory

The Journal of Combinatorial Theory, Series A and Series B, are mathematical journals specializing in combinatorics and related areas.

## K-vertex-connected graph

In graph theory, a connected graph G is said to be k-vertex-connected (or k-connected) if it has more than k vertices and remains connected whenever fewer than k vertices are removed.

## Lévy family of graphs

In graph theory, a branch of mathematics, a Lévy family of graphs is a family of graphs Gn, n.

## Levi graph

In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure.

## Longest path problem

In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph.

## Matching (graph theory)

In the mathematical discipline of graph theory, a matching or independent edge set in a graph is a set of edges without common vertices.

## Mathematical induction

Mathematical induction is a method of mathematical proof typically used to establish a given statement for all natural numbers.

## Möbius configuration

In geometry, the Möbius configuration or Möbius tetrads is a certain configuration in Euclidean space or projective space, consisting of two mutually inscribed tetrahedra: each vertex of one tetrahedron lies on a face plane of the other tetrahedron and vice versa.

## Median graph

In graph theory, a division of mathematics, a median graph is an undirected graph in which every three vertices a, b, and c have a unique median: a vertex m(a,b,c) that belongs to shortest paths between each pair of a, b, and c. The concept of median graphs has long been studied, for instance by or (more explicitly) by, but the first paper to call them "median graphs" appears to be.

## N-skeleton

In mathematics, particularly in algebraic topology, the of a topological space X presented as a simplicial complex (resp. CW complex) refers to the subspace Xn that is the union of the simplices of X (resp. cells of X) of dimensions In other words, given an inductive definition of a complex, the is obtained by stopping at the.

## Network topology

Network topology is the arrangement of the various elements (links, nodes, etc.) of a computer network.

## Pancyclic graph

In the mathematical study of graph theory, a pancyclic graph is a directed graph or undirected graph that contains cycles of all possible lengths from three up to the number of vertices in the graph.

## Partial cube

In graph theory, a partial cube is a graph that is an isometric subgraph of a hypercube.

## Path (graph theory)

In graph theory, a path in a graph is a finite or infinite sequence of edges which connect a sequence of vertices which, by most definitions, are all distinct from one another.

## Permutation

In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting.

## Planar graph

In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints.

## 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.

## Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

## Snake-in-the-box

The snake-in-the-box problem in graph theory and computer science deals with finding a certain kind of path along the edges of a hypercube.

## Spanning tree

In the mathematical field of graph theory, a spanning tree T of an undirected graph G is a subgraph that includes all of the vertices of G that is a tree.

## Subset

In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

## Symmetric graph

In the mathematical field of graph theory, a graph G is symmetric (or arc-transitive) if, given any two pairs of adjacent vertices u1—v1 and u2—v2 of G, there is an automorphism such that In other words, a graph is symmetric if its automorphism group acts transitively upon ordered pairs of adjacent vertices (that is, upon edges considered as having a direction).

## Szymanski's conjecture

In mathematics, Szymanski's conjecture, named after, states that every permutation on the n-dimensional doubly directed hypercube graph can be routed with edge-disjoint paths.

## Unit distance graph

In mathematics, and particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting two points by an edge whenever the distance between the two points is exactly one.

## Unit vector

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.

## Vertex (geometry)

In geometry, a vertex (plural vertices) is a special kind of point that describes the corners or intersections of geometric shapes.

## Vertex (graph theory)

In mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices).

## Wreath product

In mathematics, the wreath product of group theory is a specialized product of two groups, based on a semidirect product.

## References

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