Graph presentations At first sight a diagrammatic presentation of a graph and a presentation via formulas look quite different and seem to be two somehow orthogonal kinds of presentation: the former "analogical" (each node represented exactly once), the latter "symbolical" (each node represented recurrently).
I find it astonishing that — brought into an appropriate canonical form, respectively — the two presentations are nevertheless essentially the same.
Consider the graph below a) in a diagrammatic and b) in a presentation as a set of literals (atomic sentences or negation of atomic sentences).

Choose as a canonical arrangement a) an equidistant arrangement of the nodes along a straight line and the arrows as oriented semicircles, resp. b) a lexical sorting of the $3 \times 3$ sentences and arrangement into a square:

Now replace a) every present arrow by 1 and all nodes and non-present arrows by 0, resp. b) all atomic sentences by 1 and all negations by 0:

This is — essentially — the adjacency matrix of the graph, but derived in two seemingly independent and unrelated ways.
I like this analogy, but I cannot make real sense out of it. Is it just prestidigitation and hocus-pocus? Or is there something deeper behind it?

Of which "deeper" principle (or whatever) is this just a simple but vivid
  example, eventually?

 A: This is not a magic at all. 
The secret is that you can alway represent graphs here and later, i mean your type of graphs, which is simple oriented graphs) as a collection of edges with information about nodes at the start and end of the edge, and, in equivalent form, as a collection of nodes with information about inbound and ougoing edges (you have to identify it, for example, as you did - by colors,- or by any other identification method).
In my opinion, literal representation is too formal.
Lets do some magic with any of the graph, Let's prove that these representations are equal. By providing a one to one correspondence between these two representation, whe can prove that for the single oreinted graphs.
Let's make a one to one correspondence betwee two set: a set of graphs (single oriented) which can be drawn using graph visualization tools nad a set of graphs which can be built using literal notation.
Doing so we can proof that there is a bection (a function, or an reversible algorighm, which can fing a graph on a destination set, from a source graph on anbother set) between graphs.
Lets introduce the equality operator for the corresponding sets of graphs.
For the set of literally geven graphs (say, G), graphs a and b are considered equal, if the have exacly the same written rule sets.
For the set of the visually drawn graphs (say, F), graphs a and b are considered equal, if they have exacly the same circled numbers, equal sets of circle labels, and each circle has the same number of inboud and outgoing connections for each circle.
At first, let's provide a biection:
Forward direction $f(x)$: (from a drawn graph to a literally written):
Take a black marker, and Write an R letter at the left of any graph circle's letter, which has an outgoing connection), then write a letter at the right of any graph circle's letter for any graph node, which has an inbound connection.
Write down on another piece of paper, all the resulting letter triples, then, for each graph node, write down all texts in form of ~Rab, where a is a current letter, and b is any other letter which has no the direct inbound connection from the current node. Doing so we will get an exactly the same set of rules in form of {Rab|~Rab} which can be found in another graph set.
Backward $g(x)$ (from a literally written graph to a drawn one): Use a rubber an erase all triples on a paper with ~ at the begining, then for all rules in form of Rab, do the following, if a first letter after the R is not labeled and circled on another piece of paper, than place a letter on that paper an circle it, do the same for the next letter on a triple, then, connect first circle found to a last circle found with a edge with an arrow, pointing to the second labeled circle. Repeat the same procedure for all remaining rules.  You will get a graph from a first graphs' sets.
By the definition of biection functions,
$f(g(a))=a$, $g(f(b))=b$ for any a in G, b in F.
$f(x), g(x)$ is a biection between $F$ and $G$
By the construction of a biection (we did not apply any of restrictions to a graph in F or G), sets F and G is considered equal, so sets F and G is equivalent represenations of any simple oriented graph. That means you can use the both represenatons as they are considered equal!
You can choose any represenation, but i want to add, that in literal represnatation all ~Rab form rules are excessive in terms of logic, (we can always reconstruct full set of missing rules by providing simple reconstruciton function mechanism).
Moreover, the same is are true for both finite or infinite graphs, one-edged or having multiple edges, oriented or not oriented, finite and infinite etc.!
If you have a rule to draw an infinite graph, you can always provide a rule to generate a literals for a sequence of nodes in literal representaion and vice versa!
