Given the expression $E$, is there an assignment of boolean values ($true$ or $false$) that we can give to our variables such that this is evaluated to $true$?
$E = (¬x + z + ¬v) · (¬v + w) ·(¬z + w +v)$
Also, i would like to know how to associate it a graph $G$ to the boolean expression with those values that make the expression $true$.
Let $\lnot v = T \iff v = F$.
Then the first two factors are true, and for the third factor, it will be true if you put $z = F \iff \lnot z = T$ and/or put $w = T$.
Only one of $\lnot z$ or $w$ needs to be true for the third factor to be true. So the truth of your statement $E$ depends on as few as only two truth-value assignments:
$$(v = F \land z = F)\;\text{ or }\;(v = F \land w = T).$$
In each case, the truth value assignments of the two remaining variables is irrelevant: the two remaining elements can be true or false without changing the truth of the statement.
If you'd like to see all the possible truth value assignments for the variables that make $E$ true, Wolfram Alpha helped to create the following truth-table:
In the right-most column, for each row that evaluates to true (T) (they're highlighted in an awful shade of pink!), simply read off the 9 possible truth-value assignments for the variables associated with that row.