I discussed endographs of real functions in my post Endographs and cographs of real functions. Endographs of finite functions also provide another way of thinking about functions, and I show some examples here. This is not a new idea; endographs have appeared from time to time in textbooks, but they are not used much, and they have the advantage of revealing some properties of a function instantly that cannot be seen so easily in a traditional graph or cograph.
In contrast to endographs of functions on the real line, an endograph of a finite function from a set to itself contains all the information about the function. For real functions, only some of the arrows can be shown; you are dependent on continuity to interpolate where the infinite number of intermediate arrows would be, and of course, it is easy to produce a function, with, say, small-scale periodicity, that the arrows would miss, so to speak. But with an endograph of a finite function, WYSIATI (what you see is all there is).
Here is the endograph of a function. It is one function. The graph has four connected components.
You can see immediately that it is a permutation of the set , and that it is involution (a permutation for which ). In cycle notation, it is the permutation , and the connected components of the endograph correspond to the cycle structure.
Here is another permutation:
You can see that to get you would have to have , since you have to apply the 3-cycle 3 times and the transposition twice to get the identity. The cycle structure tells you this, but you have to visualize it acting to see that. The endograph gives the newbie a jumpstart on the visualization. “The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols” (Brett Victor). This is an argument for insisting that this permutation is the endograph, and the abstract string of symbols is a representation of secondary importance. [See Note 1.]
Here is the cograph of the same function. It requires a bit of visualization or tracing arrows around to see its cycle structure.
If I had rearranged the nodes like this
the cycle structure would be easier to see. This does not indicate as much superiority of the endograph metaphor over the cograph metaphor as you might think: My endograph code [Note 2] uses Mathematica’s graph-displaying algorithm, which automatically shows cycles clearly. The cograph code that I wrote specifies the placement of the nodes explicitly, so I rearranged them to obtain the second cograph above using my knowledge of the cycle structure.
The following endographs of functions that are not permutations exhibit the general fact that the graph of a finite function consists of cycles with trees attached. This structure is obvious from the endographs, and it is easy to come up with a proof of this property of finite functions by tracing your finger around the endographs.
This is the endograph of the polynomial over the finite field of 11 elements.
Quite a bit is known about polynomials over finite fields that give permutations. For example there is an easy proof using interpolating polynomials that a polynomial that gives a transposition must have degree . The best reference for this stuff is Lidl and Niederreiter, Introduction to Finite Fields and their Applications
The endographs above raise questions such as what can you say about the degree or coefficients of a polynomial that gives a digraph like the function below that is idempotent (). Students find idempotence vs. involution difficult to distinguish between. Digraphs show you almost immediately what is going on. Stare at the digraph below for a bit and you will see that if you follow to a node and then follow it again you stay where you are (the function is the identity on its image). That’s another example of the insights you can get from a new metaphor for a mathematical object.
 Atish Bagchi and I have contributed to this goal in Graph Based Logic and Sketches, which gives a bare glimpse of the possibility of considering that the real objects of logic are diagrams and their limits and morphisms between them, rather than hard-to-parse strings of letters and logical symbols. Implementing this (and implementing Brett Victor’s ideas) will require sophisticated computer support. But that support is coming into existence. We won’t have to live with string-based math forever.
 The Mathematica notebook used to produce these pictures is here. It has lots of other examples.