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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).

enter image description here

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:

enter image description here

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:

enter image description here

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?

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    Perhaps simply of the fact that the point $\left(\frac{i+j}2, \frac{j-i}2\right)$ is a 45° rotation (up to scaling) of the point $(i,j)$?2011-09-30

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