Search results
Results from the WOW.Com Content Network
In graph theory, a loop (also called a self-loop or a buckle) is an edge that connects a vertex to itself. A simple graph contains no loops. Depending on the context, a graph or a multigraph may be defined so as to either allow or disallow the presence of loops (often in concert with allowing or disallowing multiple edges between the same ...
Diagrams with loops (in graph theory, these kinds of loops are called cycles, while the word loop is an edge connecting a vertex with itself) correspond to the quantum corrections to the classical field theory. Because one-loop diagrams only contain one cycle, they express the next-to-classical contributions called the semiclassical contributions.
In graph theory, reachability refers to the ability to get from one vertex to another within a graph. A vertex s {\displaystyle s} can reach a vertex t {\displaystyle t} (and t {\displaystyle t} is reachable from s {\displaystyle s} ) if there exists a sequence of adjacent vertices (i.e. a walk ) which starts with s {\displaystyle s} and ends ...
The transitive closure of a given directed graph is a graph on the same vertex set that has an edge from one vertex to another whenever the original graph has a path connecting the same two vertices. A transitive reduction of a graph is a minimal graph having the same transitive closure; directed acyclic graphs have a unique transitive reduction.
A graph with 6 vertices and 7 edges where the vertex number 6 on the far-left is a leaf vertex or a pendant vertex. In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph ...
The transitive reduction of a finite directed graph G is a graph with the fewest possible edges that has the same reachability relation as the original graph. That is, if there is a path from a vertex x to a vertex y in graph G, there must also be a path from x to y in the transitive reduction of G, and vice versa. Specifically, if there is ...
Graph homomorphisms also form a category, with graphs as objects and homomorphisms as arrows. [37] The initial object is the empty graph, while the terminal object is the graph with one vertex and one loop at that vertex. The tensor product of graphs is the category-theoretic product and the exponential graph is the exponential object for this ...
A directed graph is strongly connected if and only if it has an ear decomposition, a partition of the edges into a sequence of directed paths and cycles such that the first subgraph in the sequence is a cycle, and each subsequent subgraph is either a cycle sharing one vertex with previous subgraphs, or a path sharing its two endpoints with ...