Graph databaseA graph database (GDB) is a database that uses graph structures for semantic queries with nodes, edges, and properties to represent and store data. A key concept of the system is the graph (or edge or relationship). The graph relates the data items in the store to a collection of nodes and edges, the edges representing the relationships between the nodes. The relationships allow data in the store to be linked together directly and, in many cases, retrieved with one operation.
Multiple edgesIn graph theory, multiple edges (also called parallel edges or a multi-edge), are, in an undirected graph, two or more edges that are incident to the same two vertices, or in a directed graph, two or more edges with both the same tail vertex and the same head vertex. A simple graph has no multiple edges and no loops. Depending on the context, a graph may be defined so as to either allow or disallow the presence of multiple edges (often in concert with allowing or disallowing loops): Where graphs are defined so as to allow multiple edges and loops, a graph without loops or multiple edges is often distinguished from other graphs by calling it a simple graph.
Strong perfect graph theoremIn graph theory, the strong perfect graph theorem is a forbidden graph characterization of the perfect graphs as being exactly the graphs that have neither odd holes (odd-length induced cycles of length at least 5) nor odd antiholes (complements of odd holes). It was conjectured by Claude Berge in 1961. A proof by Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin Thomas was announced in 2002 and published by them in 2006.
Ramanujan graphIn the mathematical field of spectral graph theory, a Ramanujan graph is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are excellent spectral expanders. As Murty's survey paper notes, Ramanujan graphs "fuse diverse branches of pure mathematics, namely, number theory, representation theory, and algebraic geometry". These graphs are indirectly named after Srinivasa Ramanujan; their name comes from the Ramanujan–Petersson conjecture, which was used in a construction of some of these graphs.
Hall's marriage theoremIn mathematics, Hall's marriage theorem, proved by , is a theorem with two equivalent formulations. In each case, the theorem gives a necessary and sufficient condition for an object to exist: The combinatorial formulation answers whether a finite collection of sets has a transversal—that is, whether an element can be chosen from each set without repetition. Hall's condition is that for any group of sets from the collection, the total unique elements they contain is at least as large as the number of sets in the group.
Ring of integersIn mathematics, the ring of integers of an algebraic number field is the ring of all algebraic integers contained in . An algebraic integer is a root of a monic polynomial with integer coefficients: . This ring is often denoted by or . Since any integer belongs to and is an integral element of , the ring is always a subring of . The ring of integers is the simplest possible ring of integers. Namely, where is the field of rational numbers. And indeed, in algebraic number theory the elements of are often called the "rational integers" because of this.
