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 order degree $2$ is complete.
Proof. All models of uncountable size $\kappa$ consist of $\kappa$ many $Z$ \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 order degree$2$have the same first order theory. In particular, the graph consisting of one$Z$-chain \mathbb{Z}$-chain is elementary equivalent to the graph consisting of any number of such $Z$ \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. 2 added 11 characters in body Stefan's original idea is realized in the following observation. Theorem. The theory of nontrivial cycle-free graphs where every vertex has order$2$is complete. Proof. All models of uncountable size$\kappa$consist of$\kappa$many$Z$chains, and hence are isomorphic. Thus, the theory is$\kappa$-categorical, and hence complete. QED Thus, all cycle-free graphs with every vertex of order$2$have the same first order theory. In particular, the graph consisting of one$Z$-chain is elementary equivalent to the graph consisting of any number of such$Z$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. 1 Stefan's original idea is realized in the following observation. Theorem. The theory of cycle-free graphs where every vertex has order$2$is complete. Proof. All models of uncountable size$\kappa$consist of$\kappa$many$Z$chains, and hence are isomorphic. Thus, the theory is$\kappa$-categorical, and hence complete. QED Thus, all cycle-free graphs with every vertex of order$2$have the same first order theory. In particular, the graph consisting of one$Z$-chain is elementary equivalent to the graph consisting of any number of such$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.