Let $L = \{R\}$ be a predicate language, with the $R$ being the relation that $2$ non-equal vertices are connected. I would like to formulate the following statements, with $v,u$ being vertices in $G$.
(a) Every vertex has at least one neighbor. My approach:
$\phi_a = (\forall v \exists u (\lnot( u\doteq v) \land (Ruv \land \lnot Rvv$))
(b) There exists one vertex with exactly two neighbors. My approach:
$\phi_b = (\exists v(\lnot((v \doteq u_1) \land (v \doteq u_2)))\land (Rvu_1 \land Rvu_2 \land \lnot Rvv))$
(c) Not all vertices have neighbors. My approach:
$\phi_c = \exists v \lnot \phi_a$
I am sure there is a lot going wrong, I am new to predicate logic. But any hints or errors would be of great help!