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The graph shown here appears as a subgraph of an undirected graph if and only if models the sentence ,,,.. In the first-order logic of graphs, a graph property is expressed as a quantified logical sentence whose variables represent graph vertices , with predicates for equality and adjacency testing.
Two-dimensional linear inequalities are expressions in two variables of the form: + < +, where the inequalities may either be strict or not. The solution set of such an inequality can be graphically represented by a half-plane (all the points on one "side" of a fixed line) in the Euclidean plane. [2]
The relation not greater than can also be represented by , the symbol for "greater than" bisected by a slash, "not". The same is true for not less than , a ≮ b . {\displaystyle a\nless b.} The notation a ≠ b means that a is not equal to b ; this inequation sometimes is considered a form of strict inequality. [ 4 ]
In graph theory, a perfect graph is a graph in which the chromatic number equals the size of the maximum clique, both in the graph itself and in every induced subgraph. In all graphs, the chromatic number is greater than or equal to the size of the maximum clique, but they can be far apart.
Ore's theorem is a result in graph theory proved in 1960 by Norwegian mathematician Øystein Ore. It gives a sufficient condition for a graph to be Hamiltonian , essentially stating that a graph with sufficiently many edges must contain a Hamilton cycle .
A drawing of a graph with 6 vertices and 7 edges.. In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.
The strong perfect graph theorem of Chudnovsky et al. (2006) states that a graph is perfect if and only if none of its induced subgraphs are cycles of odd length greater than or equal to five, or their complements. Because this characterization is unaffected by graph complementation, it immediately implies the weak perfect graph theorem.
Spectral graph theory is the branch of graph theory that uses spectra to analyze graphs. See also spectral expansion. split 1. A split graph is a graph whose vertices can be partitioned into a clique and an independent set. A related class of graphs, the double split graphs, are used in the proof of the strong perfect graph theorem.