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Independence Complex
The independence complex of a graph is a mathematical object describing the independent sets of the graph. Formally, the independence complex of an undirected graph ''G'', denoted by I(''G''), is an abstract simplicial complex (that is, a family of finite sets closed under the operation of taking subsets), formed by the sets of vertices in the independent sets of ''G''. Any subset of an independent set is itself an independent set, so I(''G'') is indeed closed under taking subsets. Every independent set in a graph is a clique in its complement graph, and vice versa. Therefore, the independence complex of a graph equals the clique complex of its complement graph, and vice versa. Homology groups Several authors studied the relations between the properties of a graph ''G'' = (''V'', ''E''), and the homology groups of its independence complex I(''G''). In particular, several properties related to the dominating sets in ''G'' guarantee that some reduced homology groups of I(''G'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Induced Matching
In graph theory, an induced matching or strong matching is a subset of the edges of an undirected graph that do not share any vertices (it is a Matching (graph theory), matching) and these are the only edges connecting any two vertices which are endpoints of the matching edges (it is an induced subgraph). An induced matching can also be described as an Independent set (graph theory), independent set in the Graph power, square of the line graph of the given graph. Strong coloring and neighborhoods The minimum number of induced matchings into which the edges of a graph can be partitioned is called its ''strong chromatic index'', by analogy with the chromatic index of the graph, the minimum number of matchings into which its edges can be partitioned. It equals the chromatic number of the square of the line graph. Brooks' theorem, applied to the square of the line graph, shows that the strong chromatic index is at most quadratic in the maximum degree of the given graph, but better ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Sets
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, * a 0-dimensional simplex is a point (mathematics), point, * a 1-dimensional simplex is a line segment, * a 2-dimensional simplex is a triangle, * a 3-dimensional simplex is a tetrahedron, and * a Four-dimensional space, 4-dimensional simplex is a 5-cell. Specifically, a -simplex is a -dimensional polytope that is the convex hull of its Vertex (geometry), vertices. More formally, suppose the points u_0, \dots, u_k are affinely independent, which means that the vectors u_1 - u_0,\dots, u_k-u_0 are linearly independent. Then, the simplex determined by them is the set of points C = \left\. A regular simplex is a simplex that is also a regular polytope. A regular -simplex may be constructed from a regular -simplex by connecti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rainbow-independent Set
In graph theory, a rainbow-independent set (ISR) is an independent set in a graph, in which each vertex has a different color. Formally, let be a graph, and suppose vertex set is partitioned into subsets , called "colors". A set of vertices is called a rainbow-independent set if it satisfies both the following conditions: * It is an independent set – every two vertices in are not adjacent (there is no edge between them); * It is a rainbow set – contains at most a single vertex from each color . Other terms used in the literature are independent set of representatives, independent transversal, and independent system of representatives. As an example application, consider a faculty with departments, where some faculty members dislike each other. The dean wants to construct a committee with members, one member per department, but without any pair of members who dislike each other. This problem can be presented as finding an ISR in a graph in which the nodes are the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Israel Journal Of Mathematics
'' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem ( Magnes Press). History Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the journal publishes articles on all areas of mathematics. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.70, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... was 0.754. External links * Mathematics journals Academic journals established in 1963 Academic journals of Israel English-language journals Bimonthly journals Hebrew University of Jerusalem {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rook (chess)
The rook (; ♖, ♜) is a piece in the game of chess. It may move any number of squares horizontally or vertically without jumping, and it may an enemy piece on its path; it may participate in castling. Each player starts the game with two rooks, one in each corner on their side of the board. Formerly, the rook (from ) was alternatively called the ''tower'', ''marquess'', ''rector'', and ''comes'' (''count'' or ''earl''). The term "castle" is considered to be informal or old-fashioned. Placement and movement The white rooks start on the squares a1 and h1, while the black rooks start on a8 and h8. The rook moves horizontally or vertically, through any number of unoccupied squares. The rook cannot jump over pieces. The rook may capture an enemy piece by moving to the square on which the enemy piece stands, removing it from play. The rook also participates with the king in a special move called castling, wherein it is transferred to the square crossed by the king after th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chessboard
A chessboard is a game board used to play chess. It consists of 64 squares, 8 rows by 8 columns, on which the chess pieces are placed. It is square in shape and uses two colours of squares, one light and one dark, in a chequered pattern. During play, the board is oriented such that each player's near-right corner square is a light square. The columns of a chessboard are known as ', the rows are known as ', and the lines of adjoining same-coloured squares (each running from one edge of the board to an adjacent edge) are known as '. Each square of the board is named using algebraic, descriptive, or numeric chess notation; algebraic notation is the FIDE standard. In algebraic notation, using White's perspective, files are labeled ''a'' through ''h'' from left to right, and ranks are labeled ''1'' through ''8'' from bottom to top; each square is identified by the file and rank which it occupies. The a- through d-files constitute the , and the e- through h-files constitute the ; the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theory), independent sets U and V, that is, every edge (graph theory), edge connects a Vertex (graph theory), vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycle (graph theory), cycles. The two sets U and V may be thought of as a graph coloring, coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a Gallery of named graphs, triangle: after one node is colored blue and another red, the third vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Graph
In the mathematics, mathematical discipline of graph theory, the line graph of an undirected graph is another graph that represents the adjacencies between edge (graph theory), edges of . is constructed in the following way: for each edge in , make a vertex in ; for every two edges in that have a vertex in common, make an edge between their corresponding vertices in . The name ''line graph'' comes from a paper by although both and used the construction before this. Other terms used for the line graph include the covering graph, the derivative, the edge-to-vertex dual, the conjugate, the representative graph, and the θ-obrazom, as well as the edge graph, the interchange graph, the adjoint graph, and the derived graph., p. 71. proved that with one exceptional case the structure of a connected graph can be recovered completely from its line graph. Many other properties of line graphs follow by translating the properties of the underlying graph from vertices into edges ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matching (graph Theory)
In the mathematical discipline of graph theory, a matching or independent edge set in an undirected Graph (discrete mathematics), graph is a set of Edge (graph theory), edges without common vertex (graph theory), vertices. In other words, a subset of the edges is a matching if each vertex appears in at most one edge of that matching. Finding a matching in a bipartite graph can be treated as a Flow network, network flow problem. Definitions Given a Graph (discrete mathematics), graph a matching ''M'' in ''G'' is a set of pairwise non-adjacent edges, none of which are loop (graph theory), loops; that is, no two edges share common vertices. A vertex is matched (or saturated) if it is an endpoint of one of the edges in the matching. Otherwise the vertex is unmatched (or unsaturated). A maximal matching is a matching ''M'' of a graph ''G'' that is not a subset of any other matching. A matching ''M'' of a graph ''G'' is maximal if every edge in ''G'' has a non-empty intersectio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological Connectivity
In algebraic topology, homological connectivity is a property describing a topological space based on its homology groups. Definitions Background ''X'' is ''homologically-connected'' if its 0-th homology group equals Z, i.e. H_0(X)\cong \mathbb, or equivalently, its 0-th reduced homology group is trivial: \tilde(X)\cong 0. * For example, when ''X'' is a graph and its set of connected components is ''C'', H_0(X)\cong \mathbb^ and \tilde(X)\cong \mathbb^ (see graph homology). Therefore, homological connectivity is equivalent to the graph having a single connected component, which is equivalent to graph connectivity. It is similar to the notion of a connected space. ''X'' is ''homologically 1-connected'' if it is homologically-connected, and additionally, its 1-th homology group is trivial, i.e. H_1(X)\cong 0. * For example, when ''X'' is a connected graph with vertex-set ''V'' and edge-set ''E'', H_1(X) \cong \mathbb^. Therefore, homological 1-connectivity is equivalent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meshulam's Game
In graph theory, Meshulam's game is a game used to explain a theorem of Roy Meshulam related to the homological connectivity of the independence complex of a graph, which is the smallest index ''k'' such that all reduced homological groups up to and including ''k'' are trivial. The formulation of this theorem as a game is due to Aharoni, Berger and Ziv. Description The game-board is a graph ''G.'' It is a zero-sum game for two players, CON and NON. CON wants to show that I(''G''), the independence complex of ''G'', has a high connectivity; NON wants to prove the opposite. At his turn, CON chooses an edge ''e'' from the remaining graph. NON then chooses one of two options: * ''Disconnection'' – remove the edge ''e'' from the graph. * ''Explosion'' – remove both endpoints of ''e'', together with all their neighbors and the edges incident to them. The score of CON is defined as follows: * If at some point the remaining graph has an isolated vertex, the score is infinity; * O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
