I am working on updating my draft paper on a general concept of means. Preprint Defining an Abstract Mean
One of the core failures of the drafts to date has been a lack of examples. There are plenty of examples which are in many ways just the arithmetic mean in disguise, but I have finally stumbled across an example that is not so.
Consider the set G = {x | x is an undirected graph, with potentially countably infinitely many points, and x^n = x, for all n in Z+} where multiplication is the Cartesian product of graphs. The set ignores duplicates which are simply isomorphisms of each other. The mean under Cartesian product exists and I'm pretty sure it is unique.
The question I have is how many elements are in G? I have a few examples already: a single point, countably infinitely many disconnected points, the infinite dimensional hypercube graph, and I believe the complete graph with countably infinitely many points. What other graphs fit? And is the set finite or infinite?
Of course any answers that I get that yield results will be cited in the final draft of the paper. I'm also open to a graph theorist that would want to work with me, though I understand that this result is unlikely given that I have not published in any peer reviewed journals yet.
There is a lot of information about means in this entry --
https://en.wikipedia.org/wiki/Generalized_mean
Hello Ross La Haye, the concept described on Wikipedia is actually less general than the one that I have devised in my paper, and is not compatible with defining a mean on these kinds of graphs. I am simply looking for information on the nature of the set of graphs that was defined above.
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