The (usual) language of graphs has only one non-logical symbol, a binary relation E (for edge). Many properties of graphs are not expressible as a first order sentence in this language. For example, it is not possible to express in a single sentence that a graph has finitely many vertices. I also believe that it is not possible to express in a single sentence that a graph is 2-colorable or that a graph is planar.
Is there an interesting fragment of graph theory that is fully or partially captured by first order logic with this language? Are there important theorems that can be expressed in first order logic with this language? (If so, then the completeness of first order logic implies that their proofs can be carried out in first order logic.) Are there interesting open questions that can be expressed in first order logic in this language?