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 $\tau:E\to V^{2}_{sym}$ 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

 $\xymatrix{1\ar@{-}[r]&2}$

uniquely (up to isomorphism) determines its quiver, but

 $\xymatrix{G:&1\ar@{-}[r]&2\ar@{-}[r]&3}$

does not uniquely determine its quiver. Indeed, there are exactly two nonisomorphic quivers with underlying graph $G$, namely:

 $\xymatrix{Q:&1\ar[r]&2\ar[r]&3\\ Q^{\prime}:&1\ar[r]&2&3\ar[l]}$
Title underlying graph of a quiver UnderlyingGraphOfAQuiver 2013-03-22 19:16:59 2013-03-22 19:16:59 joking (16130) joking (16130) 4 joking (16130) Definition msc 14L24