Let $G$ be a graph, which must have at least one node, i.e. the set of nodes and edges cannot be empty. Let $R$ be a symmetric and irreflexive relation for $G$ and each vertex. I would like to build a formula, such that if a graph $G$ with $n \geq 3$ nodes is a model of that formula, there exists no $n$-cycle, meaning that there is no cycle between all existing nodes $n$.
My idea was, to first express that $R$ is irreflexive and symmetric, so
$\forall a \forall b ((Rab \leftrightarrow Rba) \land \lnot Raa \land Rbb)$
and then include that there must exist one node $c$, which has only one edge. Because then, we can be certain there exists no cycle. I am not sure though if I need to include all nodes to formulate this, or if something like $\exists c \forall d$ is enough?
Any hints are appreciated!