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A finite graph is a graph in which the vertex set and the edge set are finite sets. Otherwise, it is called an infinite graph. Most commonly in graph theory it is implied that the graphs discussed are finite. If the graphs are infinite, that is usually specifically stated.
A ray, in an infinite graph, is an infinite simple path with exactly one endpoint. The ends of a graph are equivalence classes of rays. reachability The ability to get from one vertex to another within a graph. reachable Has an affirmative reachability. A vertex y is said to be reachable from a vertex x if there exists a path from x to y ...
In the mathematics of infinite graphs, an end of a graph represents, intuitively, a direction in which the graph extends to infinity. Ends may be formalized mathematically as equivalence classes of infinite paths, as havens describing strategies for pursuit–evasion games on the graph, or (in the case of locally finite graphs) as topological ends of topological spaces associated with the graph.
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. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called arcs, links or lines).
A stronger but unbalanced infinite form of Ramsey's theorem for graphs, the Erdős–Dushnik–Miller theorem, states that every infinite graph contains either a countably infinite independent set, or an infinite clique of the same cardinality as the original graph. [44]
A three-dimensional hypercube graph showing a Hamiltonian path in red, and a longest induced path in bold black. In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges).
InfiniteGraph is a distributed graph database implemented in Java and C++ and is from a class of NOSQL ("Not Only SQL") database technologies that focus on graph data structures. Developers use InfiniteGraph to find useful and often hidden relationships in highly connected, complex big data sets.
Kőnig's 1927 publication. Kőnig's lemma or Kőnig's infinity lemma is a theorem in graph theory due to the Hungarian mathematician Dénes Kőnig who published it in 1927. [1] It gives a sufficient condition for an infinite graph to have an infinitely long path.