Vertex (graph theory)

Source: Wikipedia, the free encyclopedia.
For other uses, see Vertex (disambiguation).
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

edges (unordered pairs of vertices), while a directed graph
consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another.

From the point of view of graph theory, vertices are treated as featureless and indivisible objects, although they may have additional structure depending on the application from which the graph arises; for instance, a semantic network is a graph in which the vertices represent concepts or classes of objects.

The two vertices forming an edge are said to be the endpoints of this edge, and the edge is said to be incident to the vertices. A vertex w is said to be adjacent to another vertex v if the graph contains an edge (v,w). The

neighborhood of a vertex v is an induced subgraph
of the graph, formed by all vertices adjacent to v.

Types of vertices

A small example network with 8 vertices and 10 edges.
Example network with 8 vertices (of which one is isolated) and 10 edges.

The degree of a vertex, denoted 𝛿(v) in a graph is the number of edges incident to it. An isolated vertex is a vertex with degree zero; that is, a vertex that is not an endpoint of any edge (the example image illustrates one isolated vertex).[1] A leaf vertex (also pendant vertex) is a vertex with degree one. In a directed graph, one can distinguish the outdegree (number of outgoing edges), denoted 𝛿 +(v), from the indegree (number of incoming edges), denoted 𝛿−(v); a source vertex is a vertex with indegree zero, while a sink vertex is a vertex with outdegree zero. A simplicial vertex is one whose neighbors form a clique: every two neighbors are adjacent. A universal vertex is a vertex that is adjacent to every other vertex in the graph.

A

vertex space
of a graph is a vector space having a set of basis vectors corresponding with the graph's vertices.

A graph is

adjacencies
in the graph and not based on any additional information.

Vertices in graphs are analogous to, but not the same as,

skeleton of a polyhedron forms a graph, the vertices of which are the vertices of the polyhedron, but polyhedron vertices have additional structure (their geometric location) that is not assumed to be present in graph theory. The vertex figure
of a vertex in a polyhedron is analogous to the neighborhood of a vertex in a graph.

See also

References

  1. ^ File:Small Network.png; example image of a network with 8 vertices and 10 edges
  • Gallo, Giorgio; Pallotino, Stefano (1988). "Shortest path algorithms". Annals of Operations Research. 13 (1): 1–79.
    S2CID 62752810
    .
  • Berge, Claude, Théorie des graphes et ses applications. Collection Universitaire de Mathématiques, II Dunod, Paris 1958, viii+277 pp. (English edition, Wiley 1961; Methuen & Co, New York 1962; Russian, Moscow 1961; Spanish, Mexico 1962; Roumanian, Bucharest 1969; Chinese, Shanghai 1963; Second printing of the 1962 first English edition. Dover, New York 2001)
  • ISBN 0-486-24775-9
    .
  • Biggs, Norman; Lloyd, E. H.; Wilson, Robin J. (1986).
    ISBN 0-19-853916-9
    .
  • ISBN 0-201-41033-8
    .
  • Harary, Frank; Palmer, Edgar M. (1973). Graphical enumeration. New York, Academic Press.
    ISBN 0-12-324245-2
    .

External links
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