underlying graph of a quiver
Let be a quiver, i.e. is a set of vertices, is a set of arrows and are functions which take each arrow to its source and target respectively.
Definition. An underlying graph of or graph associated with is a graph
such that , and is given by
In other words is a graph which is obtained from after forgeting the orientation of arrows. The definition of a graph used here is taken from this entry (http://planetmath.org/AlternativeDefinitionOfAMultigraph).
Note, that if we know the underlying graph of a quiver , then the information we have is not enough to reconstruct (except for a trivial case with no edges). The orientation of arrows is lost forever. In some cases it is possible to reconstruct up to an isomorphism of quivers (http://planetmath.org/MorphismsBetweenQuivers), for example graph
uniquely (up to isomorphism) determines its quiver, but
does not uniquely determine its quiver. Indeed, there are exactly two nonisomorphic quivers with underlying graph , namely:
|Title||underlying graph of a quiver|
|Date of creation||2013-03-22 19:16:59|
|Last modified on||2013-03-22 19:16:59|
|Last modified by||joking (16130)|