TITLE: Drawing (graphs) by numbers: a minimality question
QUESTION [5 upvotes]: Every simple graph $G$ can be represented ("drawn") by numbers in the following way:

Assign to each vertex $v_i$ a number $n_i$ such that all $n_i$, $n_j$ are coprime whenever $i\neq j$. Let $V$ be the set of numbers thus assigned. 
Assign to each maximal clique $C_j$ a unique prime number $p_j$ which is coprime to every number in $V$.
Assign to each vertex $v_i$ the product $N_i$ of its number $n_i$ and the prime numbers $p_k$ of the maximal cliques it belongs to.


Then $v_i$, $v_j$ are adjacent iff $N_i$
  and $N_j$ are not coprime,

i.e. there is a (maximal) clique they both belong to. Edit: It's enough to assign $n_i = 1$ when $v_i$ is not isolated and does not share all of its cliques with another vertex.
Being free in assigning the numbers $n_i$ and $p_j$ lets arise a lot of possibilites, but also the following question:

QUESTION
Can the numbers be assigned  systematically such that the greatest $N_i$
  is minimal (among all that do the job) — and if so: how?

It is obvious that the $n_i$ in the first step have to be primes for the greatest $N_i$ to be minimal. I have taken the more general approach for other - partly answered  - questions like "Can the numbers be assigned such that the set $\lbrace N_i \rbrace_{i=1,..,n}$ fulfills such-and-such conditions?"

REPLY [3 votes]: As an alternative to my earlier computational answer for particular graphs G, here is a worst-case description of the asymptotic growth of the minimum size of the integers Nj 
Let h be the sum of |V(G)| and the number of maximal cliques in G (bounded above by |E(G)|, which is saturated for bipartite graphs, whose edges are the maximal cliques).
Let c be the maximum number of maximal cliques to which a vertex v ∈ V(G) may belong. Because ph ≤ h (ln(h) + ln ln(h)), we can bound
$$ \large \Big(\min_{\small\text{weightings}} \; \max_{v \in V(G)}\;  N_v\Big) \;\;\in\;\; \mathrm O\Big(h^c \log^c(h)\Big). $$
This bound is asymptotically saturated by placing the largest prime weights on the vertex in the largest number of maximal cliques, and those maximal cliques of which it is a part, which is obviously a bad thing to do. But e.g. in Cayley graphs G, every vertex belongs to the same number of maximal cliques, this asymptotic growth cannot be avoided, as there will exist vertices v for which Nv will consist exclusively of a product of primes pt, for t bounded below by a constant fraction of h.
One can construct bipartite Cayley graphs in which the degree of each vertex is a constant fraction of n = |V(G)|. We then have h = α n for some 0 < α < 1; and h = |V(G)| + |E(G)| ∈ O(n2), so that
$$ \large \Big(\min_{\small\text{weightings}} \; \max_{v \in V(G)}\;  N_v\Big) \;\;\in\;\; \mathrm O\Big(n^{2\alpha n} \log^{\alpha n}(n)\Big) $$
for such graphs. Thus there exist graphs for which the coefficients Nj grow much more quickly than e.g. the factorial function.