Partial cubeIn graph theory, a partial cube is a graph that is isometric to a subgraph of a hypercube. In other words, a partial cube can be identified with a subgraph of a hypercube in such a way that the distance between any two vertices in the partial cube is the same as the distance between those vertices in the hypercube. Equivalently, a partial cube is a graph whose vertices can be labeled with bit strings of equal length in such a way that the distance between two vertices in the graph is equal to the Hamming distance between their labels.
Hypercube graphIn graph theory, the hypercube graph Q_n is the graph formed from the vertices and edges of an n-dimensional hypercube. For instance, the cube graph Q_3 is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. Q_n has 2^n vertices, 2^n – 1n edges, and is a regular graph with n edges touching each vertex. The hypercube graph Q_n may also be constructed by creating a vertex for each subset of an n-element set, with two vertices adjacent when their subsets differ in a single element, or by creating a vertex for each n-digit binary number, with two vertices adjacent when their binary representations differ in a single digit.
SquaregraphIn graph theory, a branch of mathematics, a squaregraph is a type of undirected graph that can be drawn in the plane in such a way that every bounded face is a quadrilateral and every vertex with three or fewer neighbors is incident to an unbounded face. The squaregraphs include as special cases trees, grid graphs, gear graphs, and the graphs of polyominos. As well as being planar graphs, squaregraphs are median graphs, meaning that for every three vertices u, v, and w there is a unique median vertex m(u,v,w) that lies on shortest paths between each pair of the three vertices.
Simplex graphIn graph theory, a branch of mathematics, the simplex graph κ(G) of an undirected graph G is itself a graph, with one node for each clique (a set of mutually adjacent vertices) in G. Two nodes of κ(G) are linked by an edge whenever the corresponding two cliques differ in the presence or absence of a single vertex. The empty set is included as one of the cliques of G that are used to form the clique graph, as is every set of one vertex and every set of two adjacent vertices.
Cartesian product of graphsIn 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.
Fibonacci cubeIn the mathematical field of graph theory, the Fibonacci cubes or Fibonacci networks are a family of undirected graphs with rich recursive properties derived from its origin in number theory. Mathematically they are similar to the hypercube graphs, but with a Fibonacci number of vertices. Fibonacci cubes were first explicitly defined in in the context of interconnection topologies for connecting parallel or distributed systems. They have also been applied in chemical graph theory.
Clique (graph theory)In the mathematical area of graph theory, a clique (ˈkliːk or ˈklɪk) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph is an induced subgraph of that is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph (the clique problem) is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied.
Glossary of graph theoryThis is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges.
Bipartite graphIn the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets and , that is, every edge connects a vertex in to one in . Vertex sets and are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets and may be thought of as a coloring of the graph with two colors: if one colors all nodes in blue, and all nodes in red, each edge has endpoints of differing colors, as is required in the graph coloring problem.
Distributive latticeIn mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets. As in the case of arbitrary lattices, one can choose to consider a distributive lattice L either as a structure of order theory or of universal algebra.
