Functions, graphs, and adjacency matrices One naively thinks of (continuous) functions as of graphs1 (lines drawn in a 2-dimensional coordinate space).
One often thinks of (countable) graphs2 (vertices connected by edges) as represented by adjacency matrices.
That's what I learned from early on, but only recently I recognized that the "drawn" graphs1 are nothing but generalized - continuous - adjacency matrices, and thus graphs1 are more or less the same as graphs2.
I'm quite sure that this is common (maybe implicit) knowledge among working mathematicians, but I wonder why I didn't learn this explicitly in any textbook on set or graph theory I've read. I would have found it enlightening.
My questions are:

Did I read my textbooks too
  superficially?
Is the analogy above (between
  graphs1 and
  graphs2) misleading?
Or is the analogy too obvious to be
  mentioned?


In my opinion, the similarity between graphs1 and graphs2 is only superficial. Both kinds of graphs can be thought of as particular subsets of certain kinds of Cartesian product ($\mathbb R \times \mathbb R$ and $V \times V$), but that's about as far as it goes.
Consider:


*

*A graph1 generalizes to higher dimensional functions $\mathbb R^m \to \mathbb R^n$ which cannot be thought of as a graph2 when $m \neq n$. A graph2 generalizes to graphs with labelled edges, multigraphs, and so on, which cannot be thought of as a graph1.

*Rearranging the order of elements in the adjacency matrix gives you the same graph2, but not the same graph1.

*Given a graph1, we care about things like injectivity, continuity, convexity, and so on, which do not correspond to useful properties of the corresponding graph2. Given a graph2, we care about things like connectivity, shortest paths, planarity, and so on, which do not correspond to useful properties of the corresponding graph1.
A simple example: The graph of $f(x) = x$ is a continuous line, but a graph where each vertex is only connected to itself is completely disconnected.
