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In a graph, multigraph, or pseudograph  , the valency of a vertex is the number of edges attached to it (note that a loop counts twice).
Synonymous with valence and degree. There are some unrelated things also called valence; there are of course many things all called degree.
For directed graphs, in- and out- are prefixed to any of the synonyms, to count incoming and outgoing edges separately.
If
 is used for the valency of vertex  , the notation  (or  on its own if there is no scope for confusion) denotes the maximum valency found in graph  . Another notation often seen is  and  for lowest and highest valency in
 respectively.
If the valency is the same number ( , say) for all its vertices,  is called regular. More specifically it is called  -valent or  -regular. Connected (components of)...
- ...0-valent graphs are edgeless vertices,
- ...1-valent graphs are pairs of vertices joined by an edge,
- ...2-valent graphs are cyclic graphs, i.e.
 -gons, of various sizes
- From
 these structures start getting more interesting. 3-valent (or trivalent) graphs are also known as cubic graphs.
A  -valent graph with  vertices has  edges.
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"valency" is owned by marijke.
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(view preamble)
| Other names: |
valence, degree |
| Also defines: |
 -valent, trivalent graph, cubic graph, regular, regular graph |
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Cross-references: sizes, connected, directed graphs, loop, edges, vertex, pseudograph, multigraph, graph
There are 65 references to this entry.
This is version 3 of valency, born on 2005-03-31, modified 2005-04-08.
Object id is 6922, canonical name is Valency.
Accessed 7853 times total.
Classification:
| AMS MSC: | 05C40 (Combinatorics :: Graph theory :: Connectivity) |
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Pending Errata and Addenda
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