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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.
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.
Each tree node contains information about a single graph vertex and its corresponding edges to ancestor nodes. A path from the root to a leaf corresponds to one single graph. Descendants of a g-trie node share a common sub-graph. Constructing a g-trie is well described in. [41] After constructing a g-trie, the counting part takes place. The ...
A connected graph is YΔY-reducible if it can be reduced to a single vertex by a sequence of steps, each of which is a ΔY- or YΔ-transform, the removal of a self-loop or multiple adjacency, the removal of a vertex with one neighbor, and the replacement of a vertex of degree two and its two neighboring edges by a single edge.
That is, it consists of vertices and edges (also called arcs), with each edge directed from one vertex to another, such that following those directions will never form a closed loop. A directed graph is a DAG if and only if it can be topologically ordered, by arranging the vertices as a linear ordering that is consistent with all edge ...
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 ...