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Berlekamp–van Lint–Seidel Graph
In graph theory, the Berlekamp–Van Lint–Seidel graph is a locally linear strongly regular graph with parameters (243,22,1,2). This means that it has 243 vertices, 22 edges per vertex (for a total of 2673 edges), exactly one shared neighbor per pair of adjacent vertices, and exactly two shared neighbors per pair of non-adjacent vertices. It was constructed by Elwyn Berlekamp, J. H. van Lint, and as the coset graph of the ternary Golay code. This graph is the Cayley graph of an abelian group. Among abelian Cayley graphs that are strongly regular and have the last two parameters differing by one, it is the only graph that is not a Paley graph. It is also an integral graph, meaning that the eigenvalues of its adjacency matrix are integers. Like the 9\times 9 Sudoku graph it is an integral abelian Cayley graph whose group elements all have order 3, one of a small number of possibilities for the orders in such a graph. There are five possible combinations of parameters for strong ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Berlekamp-van Lint-Seidel Graph
''The'' () is a grammatical article in English, denoting persons or things already mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite article in English. ''The'' is the most frequently used word in the English language; studies and analyses of texts have found it to account for seven percent of all printed English-language words. It is derived from gendered articles in Old English which combined in Middle English and now has a single form used with pronouns of any gender. The word can be used with both singular and plural nouns, and with a noun that starts with any letter. This is different from many other languages, which have different forms of the definite article for different genders or numbers. Pronunciation In most dialects, "the" is pronounced as (with the voiced dental fricative followed by a schwa) when followed by a consonant sound, and as (homophone of pronoun ''thee'') when followed by a v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Graph
In the mathematical field of graph theory, an integral graph is a graph whose adjacency matrix's spectrum consists entirely of integers. In other words, a graph is an integral graph if all of the roots of the characteristic polynomial of its adjacency matrix are integers. The notion was introduced in 1974 by Frank Harary and Allen Schwenk. Examples *The complete graph ''Kn'' is integral for all ''n''. *The only cycle graphs that are integral are C_3, C_4, and C_6. *If a graph is integral, then so is its complement graph; for instance, the complements of complete graphs, edgeless graphs, are integral. If two graphs are integral, then so is their Cartesian product and strong product; for instance, the Cartesian products of two complete graphs, the rook's graphs, are integral. Similarly, the hypercube graphs, as Cartesian products of any number of complete graphs K_2, are integral. *The line graph of an integral graph is again integral. For instance, as the line graph of K_4, the o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Combinatorial Theory
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTA Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. An electronic, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Games Graph
In graph theory, the Games graph is the largest known locally linear strongly regular graph. Its parameters as a strongly regular graph are (729,112,1,20). This means that it has 729 vertices, and 40824 edges (112 per vertex). Each edge is in a unique triangle (it is a locally linear graph) and each non-adjacent pair of vertices have exactly 20 shared neighbors. It is named after Richard A. Games, who suggested its construction in an unpublished communication and wrote about related constructions. Construction The construction of this graph involves the unique (up to symmetry) 56-point cap set (a subset of points with no three in line) in PG(5,3), the five-dimensional projective geometry over a three-element field. The six-dimensional projective geometry, PG(6,3), can be partitioned into a six-dimensional affine space AG(6,3) and a copy of PG(5,3) (the points at infinity with respect to the affine space). The Games graph has as its vertices the 729 points of the affine space AG(6, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conway's 99-graph Problem
In graph theory, Conway's 99-graph problem is an unsolved problem asking whether there exists an undirected graph with 99 vertices, in which each two adjacent vertices have exactly one common neighbor, and in which each two non-adjacent vertices have exactly two common neighbors. Equivalently, every edge should be part of a unique triangle and every non-adjacent pair should be one of the two diagonals of a unique 4-cycle. John Horton Conway offered a $1000 prize for its solution. Properties If such a graph exists, it would necessarily be a locally linear graph and a strongly regular graph with parameters (99,14,1,2). The first, third, and fourth parameters encode the statement of the problem: the graph should have 99 vertices, every pair of adjacent vertices should have 1 common neighbor, and every pair of non-adjacent vertices should have 2 common neighbors. The second parameter means that the graph is a regular graph with 14 edges per vertex. If this graph exists, it cannot hav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sudoku Graph
In the mathematics of Sudoku, the Sudoku graph is an undirected graph whose vertices represent the cells of a (blank) Sudoku puzzle and whose edges represent pairs of cells that belong to the same row, column, or block of the puzzle. The problem of solving a Sudoku puzzle can be represented as precoloring extension on this graph. It is an integral Cayley graph. Basic properties and examples On a Sudoku board of size n^2\times n^2, the Sudoku graph has n^4 vertices, each with exactly 3n^2-2n-1 neighbors. Therefore, it is a regular graph. The total number of edges is n^4(3n^2-2n-1)/2. For instance, the graph shown in the figure above, for a 4\times 4 board, has 16 vertices and 56 edges, and is 7-regular. For the most common form of Sudoku, on a 9\times 9 board, the Sudoku graph is a 20-regular graph with 81 vertices and 810 edges. The second figure shows how to count the neighbors of each cell in a 9\times 9 board. Puzzle solutions and graph coloring Each row, column, or block ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjacency Matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. If the graph is undirected (i.e. all of its edges are bidirectional), the adjacency matrix is symmetric. The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph theory. The adjacency matrix of a graph should be distinguished from its incidence matrix, a different matrix representation whose elements indicate whether vertex–edge pairs are incident or not, and its degree matrix, which contains information about the degree of each vertex. Definition For a simple graph with vertex set , the adjacency matrix is a square matrix such that its element is one when there is an edge from vertex to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paley Graph
In mathematics, Paley graphs are dense undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices. Paley graphs allow graph-theoretic tools to be applied to the number theory of quadratic residues, and have interesting properties that make them useful in graph theory more generally. Paley graphs are named after Raymond Paley. They are closely related to the Paley construction for constructing Hadamard matrices from quadratic residues . They were introduced as graphs independently by and . Sachs was interested in them for their self-complementarity properties, while Erdős and Rényi studied their symmetries. Paley digraphs are directed analogs of Paley graphs that yield antisymmetric conference matrices. They were introduced by (independently of Sachs, Erdős, and Rén ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''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 of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation \cdot that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The symbo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley Graph
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley), and uses a specified set of generators for the group. It is a central tool in combinatorial and geometric group theory. The structure and symmetry of Cayley graphs makes them particularly good candidates for constructing families of expander graphs. Definition Let G be a group and S be a generating set of G. The Cayley graph \Gamma = \Gamma(G,S) is an edge-colored directed graph constructed as follows: In his Collected Mathematical Papers 10: 403–405. * Each element g of G is assigned a vertex: the vertex set of \Gamma is identified with G. * Each element s of S is assigned a color c_s. * For every g \in G and s \in S, there is a directed edge of color c_s from the vertex corresponding to g to the one corresponding to gs. Not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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