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Second Neighborhood Problem
In mathematics, the second neighborhood problem is an unsolved problem about oriented graphs posed by Paul Seymour. Intuitively, it suggests that in a social network described by such a graph, someone will have at least as many friends-of-friends as friends. The problem is also known as the second neighborhood conjecture or Seymour’s distance two conjecture. Statement An oriented graph is a finite directed graph obtained from a simple undirected graph by assigning an orientation to each edge. Equivalently, it is a directed graph that has no self-loops, no parallel edges, and no two-edge cycles. The first neighborhood of a vertex v (also called its open neighborhood) consists of all vertices at distance one from v, and the second neighborhood of v consists of all vertices at distance two from v. These two neighborhoods form disjoint sets, neither of which contains v itself. In 1990, Paul Seymour conjectured that, in every oriented graph, there always exists at least one vertex v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oriented Graph
In graph theory, an orientation of an undirected graph is an assignment of a direction to each edge, turning the initial graph into a directed graph. Oriented graphs A directed graph is called an oriented graph if none of its pairs of vertices is linked by two symmetric edges. Among directed graphs, the oriented graphs are the ones that have no 2-cycles (that is at most one of and may be arrows of the graph). A tournament is an orientation of a complete graph. A polytree is an orientation of an undirected tree. Sumner's conjecture states that every tournament with vertices contains every polytree with vertices. The number of non-isomorphic oriented graphs with vertices (for ) is : 1, 2, 7, 42, 582, 21480, 2142288, 575016219, 415939243032, … . Tournaments are in one-to-one correspondence with complete directed graphs (graphs in which there is a directed edge in one or both directions between every pair of distinct vertices). A complete directed graph can be converted to an ... [...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, denoted by \Delta(G), and the minimum degree of a graph, denoted by \delta(G), are the maximum and minimum of its 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 entitled negative deg(v). Handshaking lemma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Douglas West (mathematician)
Douglas Brent West is a professor of graph theory at University of Illinois at Urbana-Champaign. He received his Ph.D. from Massachusetts Institute of Technology in 1978; his advisor was Daniel Kleitman. He is the "W" in G. W. Peck, a pseudonym for a group of six mathematicians that includes West.. He is the editor of the journal ''Discrete Mathematics''. Selected work Books * ''Introduction to Graph Theory'' - Second edition, Douglas B. West. Published by Prentice Hall 1996, 2001. * ''Mathematical Thinking: Problem-Solving and Proofs'' Second edition, John P D'Angelo and Douglas West. Published by Prentice Hall 1999. Research work * Spanning trees with many leaves, DJ Kleitman, DB West - SIAM Journal on Discrete Mathematics, 1991. * Class of Solutions to the Gossip Problem, Part II, DB West - Discrete Mathematics, 1982. * The interval number of a planar graph: three intervals suffice, ER Scheinerman, DB West - Journal of combinatorial theory. Series B, 1983. See also *Er ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Graph Theory
The ''Journal of Graph Theory'' is a peer-reviewed mathematics journal specializing in graph theory and related areas, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. It is published by John Wiley & Sons. The journal was established in 1977 by Frank Harary.Frank Harary a biographical sketch at the ACM site The are [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friendship Paradox
The friendship paradox is the phenomenon first observed by the sociologist Scott L. Feld in 1991 that most people have fewer friends than their friends have, on average. It can be explained as a form of sampling bias in which people with more friends are more likely to be in one's own friend group. In other words, one is less likely to be friends with someone who has very few friends. In contradiction to this, most people believe that they have more friends than their friends have. The same observation can be applied more generally to social networks defined by other relations than friendship: for instance, most people's sexual partners have had (on the average) a greater number of sexual partners than they have. The friendship paradox is an example of how network structure can significantly distort an individual's local observations. Mathematical explanation In spite of its apparently paradoxical nature, the phenomenon is real, and can be explained as a consequence of the gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle-free Graph
In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs. By Turán's theorem, the ''n''-vertex triangle-free graph with the maximum number of edges is a complete bipartite graph in which the numbers of vertices on each side of the bipartition are as equal as possible. Triangle finding problem The triangle finding problem is the problem of determining whether a graph is triangle-free or not. When the graph does contain a triangle, algorithms are often required to output three vertices which form a triangle in the graph. It is possible to test whether a graph with edges is triangle-free in time . Another approach is to find the trace of , where is the adjacency matrix of the graph. The trace is zero if and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tournament (graph Theory)
A tournament is a directed graph (digraph) obtained by assigning a direction for each edge in an undirected complete graph. That is, it is an orientation of a complete graph, or equivalently a directed graph in which every pair of distinct vertices is connected by a directed edge (often, called an arc) with any one of the two possible orientations. Many of the important properties of tournaments were first investigated by H. G. Landau in to model dominance relations in flocks of chickens. Current applications of tournaments include the study of voting theory and social choice theory among other things. The name ''tournament'' originates from such a graph's interpretation as the outcome of a round-robin tournament in which every player encounters every other player exactly once, and in which no draws occur. In the tournament digraph, the vertices correspond to the players. The edge between each pair of players is oriented from the winner to the loser. If player a beats player b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nathaniel Dean (mathematician)
Nathaniel Dean (January 9, 1956 – January 2021) was an African-American mathematician and educator who made contributions to abstract and algorithmic graph theory, as well as data visualization and parallel computing. Education Dean received his B.S. in Mathematics and Physics from Mississippi State University in 1978. He then received his M.S. in Applied Mathematics from Northeastern University in 1983. He received his Ph.D. in Mathematics from Vanderbilt University in 1987, with a doctoral thesis titled "Contractible Edges and Conjectures and Path and Cycle Numbers". Scientific career For the next eleven years, he worked at the Software Production Research Department of Bell Labs, where he would author over thirty scientific publications on graph theory, graph algorithms, parallel computing, and data visualization. In 1995 he posed a conjecture which led to progress on the second neighborhood problem, which remains open as of 2020. His work on using graph theory for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Power
In graph theory, a branch of mathematics, the th power of an undirected graph is another graph that has the same set of vertices, but in which two vertices are adjacent when their distance in is at most . Powers of graphs are referred to using terminology similar to that of exponentiation of numbers: is called the ''square'' of , is called the ''cube'' of , etc. Graph powers should be distinguished from the products of a graph with itself, which (unlike powers) generally have many more vertices than the original graph. Properties If a graph has diameter , then its -th power is the complete graph. If a graph family has bounded clique-width, then so do its -th powers for any fixed . Coloring Graph coloring on the square of a graph may be used to assign frequencies to the participants of wireless communication networks so that no two participants interfere with each other at any of their common neighbors, and to find graph drawings with high angular resolution. Both the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Seymour (mathematician)
Paul D. Seymour (born 26 July 1950) is a British mathematician known for his work in discrete mathematics, especially graph theory. He (with others) was responsible for important progress on regular matroids and totally unimodular matrices, the four colour theorem, linkless embeddings, graph minors and structure, the perfect graph conjecture, the Hadwiger conjecture, claw-free graphs, χ-boundedness, and the Erdős–Hajnal conjecture. Many of his recent papers are available from his website. Seymour is currently the Albert Baldwin Dod Professor of Mathematics at Princeton University. He won a Sloan Fellowship in 1983, and the Ostrowski Prize in 2004; and (sometimes with others) won the Fulkerson Prize in 1979, 1994, 2006 and 2009, and the Pólya Prize in 1983 and 2004. He received an honorary doctorate from the University of Waterloo in 2008, one from the Technical University of Denmark in 2013, and one from the École normale supérieure de Lyon in 2022. He was an invited ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjoint Sets
In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A collection of two or more sets is called disjoint if any two distinct sets of the collection are disjoint. Generalizations This definition of disjoint sets can be extended to a family of sets \left(A_i\right)_: the family is pairwise disjoint, or mutually disjoint if A_i \cap A_j = \varnothing whenever i \neq j. Alternatively, some authors use the term disjoint to refer to this notion as well. For families the notion of pairwise disjoint or mutually disjoint is sometimes defined in a subtly different manner, in that repeated identical members are allowed: the family is pairwise disjoint if A_i \cap A_j = \varnothing whenever A_i \neq A_j (every two ''distinct'' sets in the family are disjoint).. For example, the collection of sets is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distance (graph Theory)
In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance or shortest-path distance. Notice that there may be more than one shortest path between two vertices. If there is no path connecting the two vertices, i.e., if they belong to different connected components, then conventionally the distance is defined as infinite. In the case of a directed graph the distance between two vertices and is defined as the length of a shortest directed path from to consisting of arcs, provided at least one such path exists. Notice that, in contrast with the case of undirected graphs, does not necessarily coincide with —so it is just a quasi-metric, and it might be the case that one is defined while the other is not. Related concepts A metric space defined over a set of points in terms of distances in a graph defined over th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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