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Zig-zag Product Of Graphs
In graph theory, the zig-zag product of regular graphs G,H, denoted by G \circ H, is a binary operation which takes a large graph (G) and a small graph (H) and produces a graph that approximately inherits the size of the large one but the degree of the small one. An important property of the zig-zag product is that if H is a good expander, then the expansion of the resulting graph is only slightly worse than the expansion of G. Roughly speaking, the zig-zag product G \circ H replaces each vertex of G with a copy (cloud) of H, and connects the vertices by moving a small step (zig) inside a cloud, followed by a big step (zag) between two clouds, and finally performs another small step inside the destination cloud. More specifically, the start and endpoints for each edge are at the beginning and end of this "zig-zag-zig" process starting at the points in the replacement product of the two graphs. The zigzag product was introduced by . When the zig-zag product was first introduce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zig-zag Product
In graph theory, the zig-zag product of regular graphs G,H, denoted by G \circ H, is a Graph operations, binary operation which takes a large graph (G) and a small graph (H) and produces a graph that approximately inherits the size of the large one but the Degree (graph theory), degree of the small one. An important property of the zig-zag product is that if H is a good Expander graph, expander, then the expansion of the resulting graph is only slightly worse than the expansion of G. Roughly speaking, the zig-zag product G \circ H replaces each vertex of G with a copy (cloud) of H, and connects the vertices by moving a small step (zig) inside a cloud, followed by a big step (zag) between two clouds, and finally performs another small step inside the destination cloud. More specifically, the start and endpoints for each edge are at the beginning and end of this "zig-zag-zig" process starting at the points in the replacement product of the two graphs. The zigzag product was introd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a Set (mathematics), set of vertices (also called nodes or points); * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Graph
In graph theory, a regular graph is a Graph (discrete mathematics), graph where each Vertex (graph theory), vertex has the same number of neighbors; i.e. every vertex has the same Degree (graph theory), 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 is called a graph or regular graph of degree . Special cases Regular graphs of degree at most 2 are easy to classify: a graph consists of disconnected vertices, a graph consists of disconnected edges, and a graph consists of a disjoint union of graphs, disjoint union of cycle (graph theory), cycles and infinite chains. A graph is known as a cubic graph. A strongly regular graph is a regular graph where every adjacent pair of vertices has the same number of neighbors in common, and every non-adjacent pair of vertices has the same number of neighbors in common. The smal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Operations
In the mathematical field of graph theory, graph operations are operations which produce new graphs from initial ones. They include both unary (one input) and binary (two input) operations. Unary operations Unary operations create a new graph from a single initial graph. Elementary operations Elementary operations or editing operations, which are also known as graph edit operations, create a new graph from one initial one by a simple local change, such as addition or deletion of a vertex or of an edge, merging and splitting of vertices, edge contraction, etc. The graph edit distance between a pair of graphs is the minimum number of elementary operations required to transform one graph into the other. Advanced operations Advanced operations create a new graph from an initial one by a complex change, such as: * transpose graph; * complement graph; * line graph; * graph minor; * graph rewriting; * power of graph; * dual graph; * medial graph; * quotient graph; * Y- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree (graph Theory)
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex v is denoted \deg(v) or \deg v. The maximum degree of a graph G is denoted by \Delta(G), and is the maximum of G's vertices' degrees. The minimum degree of a graph is denoted by \delta(G), and is the minimum of G's vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A complete graph (denoted K_n, where n is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, n-1. In a signed graph, the number of positive edges connected to the vertex v is called positive deg(v) and the number of connected negative edges is enti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Definitions 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. Different formalisations of these notions give rise to different notions of expanders: ''edge expanders'', ''vertex expanders'', and ''spectral expanders'', as defined below. A disconnected graph is not an expander, since the boundary of a connected component is empty. Every connected finite graph is an expander; however, different connected graphs have different expansion parameters. The complete graph has the best expansion property, but it has largest possible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Replacement Product
In graph theory, the replacement product of two Graph (discrete mathematics), graphs is a graph product that can be used to reduce the degree (graph theory), degree of a graph while maintaining its connectivity (graph theory), connectivity. Suppose is a -regular graph and is an -regular graph with vertex set Let denote the replacement product of and . The vertex set of is the Cartesian product . For each vertex in and for each edge in , the vertex is adjacent to in . Furthermore, for each edge in , if is the th neighbor of and is the th neighbor of , the vertex is adjacent to in . If is an -regular graph, then is an -regular graph. References External links * Graph products {{Graph-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of logic gate, gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SL (complexity)
In computational complexity theory, SL (Symmetric Logspace or Sym-L) is the complexity class of problems log-space reducible to USTCON (''undirected s-t connectivity''), which is the problem of determining whether there exists a path between two vertices in an undirected graph, otherwise described as the problem of determining whether two vertices are in the same connected component. This problem is also called the undirected reachability problem. It does not matter whether many-one reducibility or Turing reducibility is used. Although originally described in terms of symmetric Turing machines, that equivalent formulation is very complex, and the reducibility definition is what is used in practice. USTCON is a special case of STCON (''directed reachability''), the problem of determining whether a directed path between two vertices in a directed graph exists, which is complete for NL. Because USTCON is SL-complete, most advances that impact USTCON have also impacted SL. Thus th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L (complexity)
In computational complexity theory, L (also known as LSPACE, LOGSPACE or DLOGSPACE) is the complexity class containing decision problems that can be solved by a deterministic Turing machine using a logarithmic amount of writable memory space. Formally, the Turing machine has two tapes, one of which encodes the input and can only be read, whereas the other tape has logarithmic size but can be written as well as read. Logarithmic space is sufficient to hold a constant number of pointers into the input and a logarithmic number of Boolean flags, and many basic logspace algorithms use the memory in this way. Complete problems and logical characterization Every non-trivial problem in L is complete under log-space reductions, so weaker reductions are required to identify meaningful notions of L-completeness, the most common being first-order reductions. A 2004 result by Omer Reingold shows that USTCON, the problem of whether there exists a path between two vertices in a given u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation Map
In mathematics, a rotation map is a function that represents an undirected edge-labeled graph, where each vertex enumerates its outgoing neighbors. Rotation maps were first introduced by Reingold, Vadhan and Wigderson (“Entropy waves, the zig-zag graph product, and new constant-degree expanders”, 2002) in order to conveniently define the zig-zag product and prove its properties. Given a vertex v and an edge label i, the rotation map returns the i'th neighbor of v and the edge label that would lead back to v. Definition For a ''D''-regular graph ''G'', the rotation map \mathrm_G : \times \rightarrow \times /math> is defined as follows: \mathrm_G (v,i)=(w,j) if the ''i'' th edge leaving ''v'' leads to ''w'', and the ''j'' th edge leaving ''w'' leads to ''v''. Basic properties From the definition we see that \mathrm_G is a permutation, and moreover \mathrm_G \circ \mathrm_G is the identity map (\mathrm_G is an involution). Special cases and properties * A rotation map i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Gap
In mathematics, the spectral gap is the difference between the moduli of the two largest eigenvalues of a matrix or operator; alternately, it is sometimes taken as the smallest non-zero eigenvalue. Various theorems relate this difference to other properties of the system. The spectral gap gets its name from the ''matrix spectrum'', that is, for a matrix, the list of its eigenvalues. It provides insight on diffusion within the graph: corresponding the spectral gap to the smallest non-zero eigenvalue, it is then the mode of the network state that shows the slowest exponential decay over time. See also * Cheeger constant (graph theory) * Cheeger constant (Riemannian geometry) * Eigengap * Spectral gap (physics) * Spectral radius ''Spectral'' is a 2016 Hungarian-American military science fiction action film co-written and directed by Nic Mathieu. Written with Ian Fried (screenwriter), Ian Fried & George Nolfi, the film stars James Badge Dale as DARPA research scientist Ma . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
