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]}