valency
In a graph, multigraph^{}, or pseudograph^{} $G$, the valency^{} of a vertex is the number of edges attached to it (note that a loop counts twice).
Synonymous with and . 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 $\rho (\mathrm{\text{v}})$ is used for the valency of vertex v, the notation $\rho (G)$ (or $\rho $ on its own if there is no scope for confusion) denotes the maximum valency found in graph $G$. Another notation often seen is $\delta (G)$ and $\mathrm{\Delta}(G)$ for lowest and highest valency in $G$ respectively.
If the valency is the same number ($\rho $, say) for all its vertices, $G$ is called regular. More specifically it is called $\rho $valent or $\rho $regular. Connected (components of)…

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…0valent graphs are edgeless vertices,

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…1valent graphs are pairs of vertices joined by an edge,

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…2valent graphs are cyclic graphs, i.e. $n$gons, of various sizes

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From $\rho \u2a7e3$ these structures start getting more interesting. 3valent (or trivalent) graphs are also known as cubic graphs.
A $\rho $valent graph with $n$ vertices has $n\rho /2$ edges.
Title  valency 

Canonical name  Valency 
Date of creation  20130322 15:10:17 
Last modified on  20130322 15:10:17 
Owner  marijke (8873) 
Last modified by  marijke (8873) 
Numerical id  6 
Author  marijke (8873) 
Entry type  Definition 
Classification  msc 05C40 
Synonym  valence 
Synonym  degree 
Defines  $\rho $valent 
Defines  trivalent graph 
Defines  cubic graph 
Defines  regular 
Defines  regular graph 