underlying graph of a quiver

Let $Q=(Q_0,Q_1,s,t)$ be a quiver, i.e. $Q_0$ is a set of vertices, $Q_1$ is a set of arrows and $s,t:Q_1\to Q_0$ are functions which take each arrow to its source and target respectively.

Definition. An underlying graph of $Q$ or graph associated with $Q$ is a graph

$$G=(V,E,\tau )$$

such that $V=Q_0$, $E=Q_1$ and $\tau :E\to V_{sym}^2$ is given by

$$\tau (\alpha )={[s(\alpha ),t(\alpha )]}_{\sim }.$$

In other words $G$ is a graph which is obtained from $Q$ 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 $G$ of a quiver $Q$, then the information we have is not enough to reconstruct $Q$ (except for a trivial case with no edges). The orientation of arrows is lost forever. In some cases it is possible to reconstruct $Q$ 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 $G$, namely:

Title	underlying graph of a quiver
Canonical name	UnderlyingGraphOfAQuiver
Date of creation	2013-03-22 19:16:59
Last modified on	2013-03-22 19:16:59
Owner	joking (16130)
Last modified by	joking (16130)
Numerical id	4
Author	joking (16130)
Entry type	Definition
Classification	msc 14L24