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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 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.
A graph edge represents a relationship between two entities, e.g. pairwise interactions or similarity based on comparisons of geometric neighborhoods (for example of pixels in images) or of another feature, with the edge weight encoding the strength of this relationship.
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
A directed graph is strongly connected or strong if it contains a directed path from x to y (and from y to x) for every pair of vertices (x, y). The strong components are the maximal strongly connected subgraphs. A connected rooted graph (or flow graph) is one where there exists a directed path to every vertex from a distinguished root vertex.
A numerical value, assigned as a label to a vertex or edge of a graph. The weight of a subgraph is the sum of the weights of the vertices or edges within that subgraph. weighted graph A graph whose vertices or edge s have been assigned weight s. A vertex-weighted graph has weights on its vertices and an edge-weighted graph has weights on its edges.
Contracting an edge without creating multiple edges. As defined below, an edge contraction operation may result in a graph with multiple edges even if the original graph was a simple graph. [2] However, some authors [3] disallow the creation of multiple edges, so that edge contractions performed on simple graphs always produce simple graphs.
In computer science, a graph is an abstract data type that is meant to implement the undirected graph and directed graph concepts from the field of graph theory within mathematics. A graph data structure consists of a finite (and possibly mutable) set of vertices (also called nodes or points ), together with a set of unordered pairs of these ...