I agree. And connectedness is not expressible in first-order logic by compactness argument.
I said no, because if there were such a formula, then there would be a formula saying that a graph is connected. What is your opinion?We define the distance between two vertices and of a graph as the least number of edges in a path from to If no such path exists, then ... is the vocabulary of graphs.
(b) Does there exist a formula so that, for any graph , if and only if ? Explain your answer.