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A graph with a loop on vertex 1. 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 ...
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 ...
A loop is an edge that joins a vertex to itself. Graphs as defined in the two definitions above cannot have loops, because a loop joining a vertex to itself is the edge (for an undirected simple graph) or is incident on (for an undirected multigraph) {,} = {} which is not in {{,},}. To allow loops, the definitions must be expanded.
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 ...
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 ...
A directed 1-forest – most commonly called a functional graph (see below), sometimes maximal directed pseudoforest – is a directed graph in which each vertex has outdegree exactly one. [8] If D is a directed pseudoforest, the undirected graph formed by removing the direction from each edge of D is an undirected pseudoforest.