Terminology First-Order Logic A graph $G$ is said to be a  model of a first-order sentence $\varphi$ if $G$ satisfies $\varphi$. Now let $\varphi(x_1,...,x_r)$ be a first order formula with free variables $x_1,...,x_r$. What is the standard terminology for a tuple of vertices $(v_1,...,v_r)$ of $G$ such that $G$ satisfies the formula $\varphi(v_1,...,v_r)$ obtained by substituting $x_i$ with $v_i$? 

What I'm having trouble is to find the right terminology to connect $G$ to the assignment. The only thing I can think now is the following.

"$(v_1,...,v_r)$ is a satisfying assignment for $\varphi$ in $G$." 

But maybe there is a shorter way.
I think that's often called a "variable assignment", since it assigns to each variable a value (a vertex in the graph, in your case).

If the graph is equipped with a set of such tuples, these might be considered to be hyperedges, i.e. edges connected to an arbitrary number of vertices (not necessarily two of them).