  This paper opens and discusses the question originally due to Daniel Herden,
who asked for which graph $(\mu,R)$ we can find a family $\{\mathbb G_\alpha:
\alpha < \mu\}$ of abelian groups such that for each $\alpha,\beta\in\mu$:
$$Ext(\mathbb G_\alpha, \mathbb G_\beta) = 0 \Longleftrightarrow(\alpha,\beta)
\in R.$$
  In this regard, we present four results. First, we give a connection to
Quillen's small object argument which helps $Ext$ vanishes and uses to present
useful criteria to the question. Suppose $\lambda = \lambda^{\aleph_0}$ and
$\mu = 2^\lambda$. We apply Jensen's diamond principle along with the criteria
to present $\lambda$-free abelian groups representing bipartite graphs.
  Third, we use a version of the black box to construct in ZFC, a family of
$\aleph_1$-free abelian groups representing bipartite graphs.
  Finally, applying forcing techniques, we present a consistent positive answer
for general graphs.
