I am trying to present a graph formula using propositional logic without any quantifiers $\forall$ and $\exists$. Is this possible?
The formula is: $$∃v_1, . . . , ∃v_n \forall u \exists v(adj(u, v) \vee u = v) \land (v = v_1 \vee · · · \vee v = v_n))$$
Suppose you have an $n$-object finite domain, where every object has a name $a_1, a_2, \ldots, a_n$. Then there is a sense in which $\forall x\varphi x$ could be traded in for $\varphi a_1 \land \varphi a_2 \land \ldots \land \varphi a_n$; and $\exists x\varphi x$ could be traded in $\varphi a_1 \lor \varphi a_2 \lor \ldots \lor \varphi a_n$. (Though even here we have to be careful -- e.g. we can't say the two wffs in each pair are logically equivalent.)
But finite-object domains are the exception, not the rule, in maths. And so in general we can't trade in quantified claims for long conjunctions/disjunctions. The use of the quantifiers is ineliminable.