 The exponential graph is the graph with all functions as vertices (not only homomorphisms) and two functions "f","g" adjacent when
In particular, there is a loop at a function "f" (it is adjacent to itself) if and only if the function gives a homomorphism from "G" to "K".Seen differently, there is an edge between "f" and "g" whenever the two functions define a homomorphism from "G" × "K"2 (the bipartite double cover of "G") to "K".The exponential graph is the exponential object in the category of graphs.