  We use results concerning the Smith forms of circulant matrices to identify
when cyclically presented groups have free abelianisation and so can be
Labelled Oriented Graph (LOG) groups. We generalize a theorem of Odoni and
Cremona to show that for a fixed defining word, whose corresponding representer
polynomial has an irreducible factor that is not cyclotomic and not equal to
$\pm t$, there are at most finitely many $n$ for which the corresponding
$n$-generator cyclically presented group has free abelianisation. We classify
when Campbell and Robertson's generalized Fibonacci groups $H(r,n,s)$ are LOG
groups and when the Sieradski groups are LOG groups. We prove that amongst
Johnson and Mawdesley's groups of Fibonacci type, the only ones that can be LOG
groups are Gilbert-Howie groups $H(n,m)$. We conjecture that if a Gilbert-Howie
group is a LOG group, then it is a Sieradski group, and prove this in certain
cases (in particular, for fixed $m$, the conjecture can only be false for
finitely many $n$). We obtain necessary conditions for a cyclically presented
group to be a connected LOG group in terms of the representer polynomial and
apply them to the Prishchepov groups.
