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I give an explicitly verifiable necessary and sufficient condition for the uniqueness of the eigenform on finitely ramified fractals, once an eigenform is known. This improves the results of my previous paper [16], where I gave some necessary and some sufficient conditions, and with a relatively mild additional requirement on the known eigenform. The result of this paper can be interpreted as a uniqueness result for self-similar energies on finitely ramified fractals. [ABSTRACT FROM AUTHOR]

I give some rather general criteria for the uniqueness of the eigenform on finitely ramified fractals. In particular, I characterize the uniqueness in the case of four vertices with some additional condition, and give some examples of non-tree symmetric fractals where the eigenform is not unique. [ABSTRACT FROM AUTHOR]