The answer is "yes": there is such a family of $F$ functions. In fact, a single computable function, acting on a single integer argument, suffices. We may do this by storing essentially complete information about the graph, and about the process of "constructing" the graph $G$ (that is, the process of computing suitable integers $n_j$ representing vertices $v_j \in V(G)$), in the integers $n_j$ themselves.

We may specify a graph on $n$ vertices using a single integer by a number of different methods, such as using binary representation to use an $n^2$ bit integer to give the adjacency matrix of the graph. (The leading 1 represents an entry in the diagonal, which may serve to define the size of the graph without denoting any edges if we consider only simple graphs; other representations than binary allow for the expression of graphs with loops.) Denote this binary representation of the graph $G$ by $g \in \mathbb N$. We compute the integers $n_j$ by carrying $g$ as the exponents of different primes, and using other prime factors to "induce" adjacency between the integers representing different integers. Let $p_j$ denote the $j$th prime, with $p_1 = 2$, $p_2 = 3$, etc. We define
$$ N_j = p_j^g $$
which will act as a "label" of sorts for our vertices; each vertex $v_j$ of $G$ (for an arbitrary ordering of the vertices) will be divisible by the prime $p_j$, but not by any other prime $p_k$ for $1 \le k \le n$. At the same time, this labelling carries with it an entire description of the graph, as well as the vertex ordering (in this case, given by the ordering of the rows/columns of the adjacency matrix of $G$).

Thus, the smallest prime factor of the integer $n_j$ corresponding to the vertex $v_j$ will indicate which vertex it is in the order, and carry a complete description of the graph $G$. Thus, we may let $F(x)$ be the function which computes the following:
<ol>
<li> Determine the smallest prime $p_{j-1} \mid x$.
<li> Determine the exponent, $g$, of the largest power of $p_{j-1}$ which divides $x$.
<li> Extract from $g$ complete information about the graph $G$, including its size $n$.
<li> Determine the next largest prime $p_j > p_{j-1}$.
<li> Compute $N_j = p_j^g$.
<li> Compute $M_j = p_{n+j}$.
<li> For each $1 \le k &lt; j$:
<ul>
<li> Compute the prime $p_{n+k}$.
<li> If $v_j$ is adjacent to $v_k$ in $G$, set $M_j \leftarrow M_j p_{n+k}$.
</ul>
<li> Return $y = N_j M_j$.
</ol>
If we define $n_1 = 2^g p_{n+1}$, this will suffice to produce an $F$-sequence of length $n$ which represents the graph $G$; each vertex $v_j$ is represented by an integer $n_j$ which is the only one amidst the first $n$ to be divisible by $2 \le p_j \le p_n$, and is adjacent to a vertex $v_k$ occurring earlier in the vertex sequence if and only if they share the common factor $p_{n+j}$.

It should be noted that these are standard tricks in computability theory; I'm effectively just adapting Godel numbering for a special purpose in this case. If you want an "interesting" way of representing graphs by models in integer sequences, which does not just amount to packing and unpacking of data structures in the integers, you're going to have to impose some non-trivial bounds --- on the integer sizes, on the computational efficiency of the procedure, etc. --- and then, you will almost certainly have to be satisfied with obtaining only some special class of graphs. But that special class may still be an interesting one, if your computational constraints are well-chosen.