Is non-connectedness of graphs first order axiomatizable?

Stefan's original idea is realized in the following observation, which shows that one $\mathbb{Z}$-chain is elementary equivalent to two such chains.

Theorem. The theory of nontrivial cycle-free graphs where every vertex has degree $2$ is complete.

Proof. All models of uncountable size $\kappa$ consist of $\kappa$ many $\mathbb{Z}$ chains, and hence are isomorphic. Thus, the theory is $\kappa$-categorical, and hence complete. QED

Thus, all cycle-free graphs with every vertex of degree $2$ have the same first order theory. In particular, the graph consisting of one $\mathbb{Z}$-chain is elementary equivalent to the graph consisting of any number of such $\mathbb{Z}$ chains. Since the first graph is connected and the latter are not, it follows that neither connectivity nor disconnectivity are first-order expressible as theories in the language of graph theory.

The class of non-connected graphs is not axiomatizable. To see this, consider $\mathbb{Z}$ as a graph with $i$, $j$ connected by an edge if and only if $|i-j|=1$. Then a simple compactness argument yields a non-connected graph $\Gamma$ such that $\Gamma$ is elementarily equivalent to $\mathbb{Z}$; ie $\Gamma$ and $\mathbb{Z}$ satisfy precisely the same first order sentences. Since $\mathbb{Z}$ is connected and $\Gamma$ is non-connected, the result follows.