Example of Exponential Graph Given a definition below (source: On Hedetniemi's Conjecture and the color template scheme by C. Tardiff and X Zhu): 
The exponential graph $G^H$ has all the functions from vertex-set of $H$ to that of $G$ as vertices and two of these functions $f,g$ are joined by an edge if $[f(u),g(v) ∈ E(G)]$ for all $[u,v] ∈ E(H)$.
Can anyone give me an example of exponential graph based on that definition?
Thank you.
 A: OK, I'll try to give a very small example with some explanation.
As soon as you understand the definition you should be able to do larger examples by yourself.
The vertices of $G^H$ are indeed functions.
They are independent of the graph structure of $G$ and $H$, they only depend on their vertex sets.
So let $G=\{0,1\}$, $H=\{a,b\}$. (I choose different sets hoping to make things clearer.)
There are four possible functions $H\to G$.
$f_0$ maps everything to 0, $f_1$ maps everything to 1, $g$ maps $a\to 0,b\to 1$
and $h$ maps $a\to 1,b\to 0$.
So our graph $G^H$ has these four functions as vertices.
Now for the edges we do need the graph structure of both $G$ and $H$.
So let us assume $G$ has only the edge $[0,1]$ and $H$ has only the edge $[a,b]$.
As an example let's ask ourselves if $[f_0,f_1]$ is an edge.
Since $H$ has only edge $[a,b]$ we just need to check if $[f_0(a),f_1(b)]=[0,1]$ is an edge of $G$,
which is true.
(Note that if you work with simple graphs, then $[b,a]$ is the same edge, but
$[f(b),g(a)]$ will generally be different from $[f(a),g(b)]$).
Now you have to do the adjacency check for all possible pairs.
How many pairs you need to check depends on whether you are talking about simple graphs or digraphs,
and whether or not you want to consider loops.
Regarding the question in your comment: in this case the result graph has 65536 vertices,
so the question "what does it look like" will not be adequately answered by providing an incidence matrix.
You may be looking for a statement like: this is isomorphic to $<$some famous graph$>$, but the argument below makes this improbable.
Another way of telling "what does it look like" may be to note that the vertices of $G^H$ are exactly the 8-digit numbers in base 4 and then characterize when two of those numbers are adjacent.
Because all vertices in the complete graph are adjacent but there are no loops (I assume),
our edge condition will be that $p_0\ldots p_7$ is adjacent to $q_0\ldots q_7$ if
and only if $p_i$ is different from $q_{i+1},q_{i+2},q_{i+6}$ and $q_{i+7}$ for all $i$,
where the $p_i$ and $q_i$ are base-4 digits and the indices are taken modulo 8.
Note that this graph has isolated points. $01123333$ will be an isolated point, since (taking $i=2$)
no base-4 digit can be different from $0,1,2$ and 3.
So our graph probably will not be isomorphic to some "famous" graph, since those tend to be connected.
