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Dominating Vertex
In graph theory, a universal vertex is a vertex of an undirected graph that is adjacent to all other vertices of the graph. It may also be called a dominating vertex, as it forms a one-element dominating set in the graph. (It is not to be confused with a universally quantified vertex in the logic of graphs.) A graph that contains a universal vertex may be called a cone. In this context, the universal vertex may also be called the apex of the cone. However, this terminology conflicts with the terminology of apex graphs, in which an apex is a vertex whose removal leaves a planar subgraph. In special families of graphs The stars are exactly the trees that have a universal vertex, and may be constructed by adding a universal vertex to an independent set. The wheel graphs, similarly, may be formed by adding a universal vertex to a cycle graph. In geometry, the three-dimensional pyramids have wheel graphs as their skeletons, and more generally the graph of any higher-dimensional pyrami ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Vertex
In graph theory, a universal vertex is a vertex of an undirected graph that is adjacent to all other vertices of the graph. It may also be called a dominating vertex, as it forms a one-element dominating set in the graph. (It is not to be confused with a universally quantified vertex in the logic of graphs.) A graph that contains a universal vertex may be called a cone. In this context, the universal vertex may also be called the apex of the cone. However, this terminology conflicts with the terminology of apex graphs, in which an apex is a vertex whose removal leaves a planar subgraph. In special families of graphs The stars are exactly the trees that have a universal vertex, and may be constructed by adding a universal vertex to an independent set. The wheel graphs, similarly, may be formed by adding a universal vertex to a cycle graph. In geometry, the three-dimensional pyramids have wheel graphs as their skeletons, and more generally the graph of any higher-dimensional pyrami ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trivially Perfect Graph
In graph theory, a trivially perfect graph is a graph with the property that in each of its induced subgraphs the size of the maximum independent set equals the number of maximal cliques. Trivially perfect graphs were first studied by but were named by ; Golumbic writes that "the name was chosen since it is trivial to show that such a graph is perfect." Trivially perfect graphs are also known as comparability graphs of trees, arborescent comparability graphs, and quasi-threshold graphs. Equivalent characterizations Trivially perfect graphs have several other equivalent characterizations: *They are the comparability graphs of order-theoretic trees. That is, let be a partial order such that for each , the set is well-ordered by the relation , and also possesses a minimum element . Then the comparability graph of is trivially perfect, and every trivially perfect graph can be formed in this way. *They are the graphs that do not have a path graph or a cycle graph as induced sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Sequence
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 T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Split Graph
In graph theory, a branch of mathematics, a split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Split graphs were first studied by , and independently introduced by . A split graph may have more than one partition into a clique and an independent set; for instance, the path is a split graph, the vertices of which can be partitioned in three different ways: #the clique and the independent set #the clique and the independent set #the clique and the independent set Split graphs can be characterized in terms of their forbidden induced subgraphs: a graph is split if and only if no induced subgraph is a cycle on four or five vertices, or a pair of disjoint edges (the complement of a 4-cycle). Relation to other graph families From the definition, split graphs are clearly closed under complementation. Another characterization of split graphs involves complementation: they are chordal graphs the complements of which are also ... [...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] |
Strong Product Of Graphs
In graph theory, the strong product is a way of combining two graphs to make a larger graph. Two vertices are adjacent in the strong product when they come from pairs of vertices in the factor graphs that are either adjacent or identical. The strong product is one of several different graph product operations that have been studied in graph theory. The strong product of any two graphs can be constructed as the union of two other products of the same two graphs, the Cartesian product of graphs and the tensor product of graphs. An example of a strong product is the king's graph, the graph of moves of a chess king on a chessboard, which can be constructed as a strong product of path graphs. Decompositions of planar graphs and related graph classes into strong products have been used as a central tool to prove many other results about these graphs. Care should be exercised when encountering the term ''strong product'' in the literature, since it has also been used to denote the tenso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost All
In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathematical context; for instance, it can mean finite, countable, or null. In contrast, "almost no" means "a negligible amount"; that is, "almost no elements of X" means "a negligible amount of elements of X". Meanings in different areas of mathematics Prevalent meaning Throughout mathematics, "almost all" is sometimes used to mean "all (elements of an infinite set) but finitely many". This use occurs in philosophy as well. Similarly, "almost all" can mean "all (elements of an uncountable set) but countably many". Examples: * Almost all positive integers are greater than 1012. * Almost all prime numbers are odd (2 is the only exception). * Almost all polyhedra are irregular (as there are only nine exceptions: the five platonic solids and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dismantlable Graph
In graph theory, a cop-win graph is an undirected graph on which the pursuer (cop) can always win a pursuit–evasion game against a robber, with the players taking alternating turns in which they can chose to move along an edge of a graph or stay put, until the cop lands on the robber's vertex. Finite cop-win graphs are also called dismantlable graphs or constructible graphs, because they can be dismantled by repeatedly removing a dominated vertex (one whose closed neighborhood is a subset of another vertex's neighborhood) or constructed by repeatedly adding such a vertex. The cop-win graphs can be recognized in polynomial time by a greedy algorithm that constructs a dismantling order. They include the chordal graphs, and the graphs that contain a universal vertex. Definitions Pursuit–evasion Cop-win graphs can be defined by a pursuit–evasion game in which two players, a cop and a robber, are positioned at different initial vertices of a given undirected graph. The cop choo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friendship Graph
In the mathematical field of graph theory, the friendship graph (or Dutch windmill graph or -fan) is a planar, undirected graph with vertices and edges. The friendship graph can be constructed by joining copies of the cycle graph with a common vertex, which becomes a universal vertex for the graph. By construction, the friendship graph is isomorphic to the windmill graph . It is unit distance with girth 3, diameter 2 and radius 1. The graph is isomorphic to the butterfly graph. Friendship theorem The friendship theorem of states that the finite graphs with the property that every two vertices have exactly one neighbor in common are exactly the friendship graphs. Informally, if a group of people has the property that every pair of people has exactly one friend in common, then there must be one person who is a friend to all the others. However, for infinite graphs, there can be many different graphs with the same cardinality that have this property. A combinatorial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friendship Theorem
In the mathematical field of graph theory, the friendship graph (or Dutch windmill graph or -fan) is a planar, undirected graph with vertices and edges. The friendship graph can be constructed by joining copies of the cycle graph with a common vertex, which becomes a universal vertex for the graph. By construction, the friendship graph is isomorphic to the windmill graph . It is unit distance with girth 3, diameter 2 and radius 1. The graph is isomorphic to the butterfly graph. Friendship theorem The friendship theorem of states that the finite graphs with the property that every two vertices have exactly one neighbor in common are exactly the friendship graphs. Informally, if a group of people has the property that every pair of people has exactly one friend in common, then there must be one person who is a friend to all the others. However, for infinite graphs, there can be many different graphs with the same cardinality that have this property. A combinatorial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Threshold Graph
In graph theory, a threshold graph is a graph that can be constructed from a one-vertex graph by repeated applications of the following two operations: # Addition of a single isolated vertex to the graph. # Addition of a single dominating vertex to the graph, i.e. a single vertex that is connected to all other vertices. For example, the graph of the figure is a threshold graph. It can be constructed by beginning with a single-vertex graph (vertex 1), and then adding black vertices as isolated vertices and red vertices as dominating vertices, in the order in which they are numbered. Threshold graphs were first introduced by . A chapter on threshold graphs appears in , and the book is devoted to them. Alternative definitions An equivalent definition is the following: a graph is a threshold graph if there are a real number S and for each vertex v a real vertex weight w(v) such that for any two vertices v,u, uv is an edge if and only if w(u)+w(v)> S. Another equivalent definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor of 0.813. Scope ''Proceedings of the American Mathematical Society'' publishes articles from all areas of pure and applied mathematics, including topology, geometry, analysis, algebra, number theory, combinatorics, logic, probability and statistics. Abstracting and indexing This journal is indexed in the following databases: 2011. American Mathematical Society. * |
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