Revision 4394875 of "ผู้ใช้:Nullzero/กระบะทราย/จุดยอด" on thwiki

{{Other uses|Vertex (disambiguation)}}

[[ไฟล์:6n-graf.svg|thumb|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]]
ใน[[ทฤษฎีกราฟ]] '''จุดยอด''' หรือ '''โหนด''' เป็นหน่วยย่อยหลักซึ่งทำให้เกิด[[กราฟ]] [[กราฟไม่ระบุทิศทาง]]ประกอบด้วยเซตของจุดยอดและเซตของเส้นเชื่อม (คู่แบบไม่มีอันดับของจุดยอด) ในขณะที่[[กราฟระบุทิศทาง]]ประกอบด้วยเซตของจุดยอดและเซตของเส้นเชื่อมที่มีทิศทาง (คู่อันดับของจุดยอด) อย่างไรก็ตาม ในทฤษฎีกราฟ จุดยอดเป็นสิ่งที่ไม่มีความน่าสนใจและแบ่งแยกไม่ได้ แม้ว่ามันอาจจะมีคุณสมบัติเพิ่มเติมขึ้นอยู่กับการใช้งาน 
<!-- for instance, a [[semantic network]] is a graph in which the vertices represent concepts or classes of objects. -->

จุดยอดสองจุดที่ทำให้เกิดเส้นเชื่อมอาจเรียกว่าเป็นจุดปลายของเส้นเชื่อม และเส้นเชื่อมนั้นก็เรียกได้ว่าเกิดจากจุดยอดเหล่านั้น <!-- แปลไม่ค่อยถูก The two vertices forming an edge are said to be the endpoints of this, and the edge is said to be incident to the vertices --> จุดยอด ''w'' เรียกว่าอยู่ประชิดกับจุดยอด ''v'' โดยที่ ''v'' ไม่ใช่ ''w'' ก็ต่อเมื่อกราฟนั้นมีเส้นเชื่อม (''v'',''w'') เพื่อนบ้านของจุดยอด ''v'' คือจุดยอดทั้งหมดที่ประชิดกับ ''v''
<!-- The [[neighborhood (graph theory)|neighborhood]] of a vertex ''v'' is an [[induced subgraph]] of the graph -->

<!--
The [[degree (graph theory)|degree]] of a vertex 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. 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) from the indegree (number of incoming edges) ; a '''source vertex''' is a vertex with indegree zero, while a '''sink vertex''' is a vertex with outdegree zero.

A [[cut vertex]] is a vertex the removal of which would disconnect the remaining graph; a [[vertex separator]] is a collection of vertices the removal of which would disconnect the remaining graph into small pieces. A [[k-vertex-connected graph]] is a graph in which removing fewer than ''k'' vertices always leaves the remaining graph connected. An [[Independent set (graph theory)|independent set]] is a set of vertices no two of which are adjacent, and a [[vertex cover]] is a set of vertices that includes the endpoint of each edge in the graph. The [[vertex space]] of a graph is a vector space having a set of basis vectors corresponding with the graph's vertices.

A graph is [[vertex-transitive graph|vertex-transitive]] if it has symmetries that map any vertex to any other vertex. In the context of [[graph enumeration]] and [[graph isomorphism]] it is important to distinguish between '''labeled vertices''' and '''unlabeled vertices'''. A labeled vertex is a vertex that is associated with extra information that enables it to be distinguished from other labeled vertices; two graphs can be considered isomorphic only if the correspondence between their vertices pairs up vertices with equal labels. An unlabeled vertex is one that can be substituted for any other vertex based only on its adjacencies in the graph and not based on any additional information.

Vertices in graphs are analogous to, but not the same as, [[vertex (geometry)|vertices of polyhedra]]: the [[skeleton (topology)|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.

In a [[directed graph]], the [[forward star]] of a vertex <math>u</math> is defined as its outgoing edges. In a Graph <math>G</math> with the set of vertices <math>V</math> and the set of edges <math>E</math>, the forward star of <math>u</math> can be described as
:<math>\{(u, v) \in E\}.\ </math><ref>{{harv|Gallo|Pallotino|1988|p=4}}</ref>
-->

== Notes ==
{{reflist|2}}

== References ==
* {{cite journal
| last = Gallo
| first = Giorgio
| authorlink = Giorgio Gallo
| last2 = Pallotino
| first2 = Stefano
| authorlink2 = Stefano Pallottino
| title = Shortest Path Algorithms 
| journal = Annals of Operations Research
| volume = 13
| issue = 1
| pages = 1–79 <!-- the inline reference refers to page 4 -->
| year = 1988
| doi = 10.1007/BF02288320
| ref = harv
  }}
* [[Claude Berge|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)
* {{Cite book | last=Chartrand | first=Gary | authorlink=Gary Chartrand | coauthors= | title=Introductory graph theory | date=1985 | publisher=Dover | location=New York  | isbn=0-486-24775-9 | pages=}}
* {{Cite book | author=Biggs, Norman; Lloyd, E. H.; Wilson, Robin J. | authorlink= | coauthors= | title=Graph theory, 1736-1936 | date=1986 | publisher=Clarendon Press | location=Oxford [Oxfordshire]  | isbn=0-19-853916-9 | pages=}}
* {{Cite book | last=Harary | first=Frank | authorlink=Frank Harary | coauthors= | title=Graph theory | date=1969 | publisher=Addison-Wesley Publishing | location=Reading, Mass.  | isbn=0-201-41033-8 | pages=}}
* {{Cite book | author=Harary, Frank; Palmer, Edgar M. | authorlink= | coauthors= | title=Graphical enumeration | date=1973 | publisher=New York, Academic Press | location= | isbn=0-12-324245-2 | pages=}}

== External links ==
* {{mathworld | title = Graph Vertex | urlname = GraphVertex}}

{{DEFAULTSORT:Vertex (Graph Theory)}}
[[หมวดหมู่:Graph theory objects]]

[[ar:رأس (نظرية المخططات)]]
[[de:Knoten (Graphentheorie)]]
[[es:Vértice (teoría de grafos)]]
[[eo:Vertico (grafeteorio)]]
[[fa:رأس (نظریه گراف)]]
[[it:Vertice (teoria dei grafi)]]
[[he:צומת (תורת הגרפים)]]
[[no:Nod]]
[[pl:Wierzchołek izolowany]]
[[pt:Vértice (teoria dos grafos)]]
[[sl:Točka (teorija grafov)]]
[[sv:Nod (grafteori)]]