Searching google for "arbitrarily small spectral gap" produces this paper: <www.mis.mpg.de/fileadmin/jjost/abj21-6-04.pdf> which claims to construct graphs of arbitrarily small spectral gap given mild conditions on the degrees of the vertices. What is your starting point? If you know each entry of $A$ then you can presumably compute the eigenvalues and hence the gap. On the other hand if you want a theorem that bounds the gap for some general $A$, then there is no hope without more assumptions on the graph.

Do you have any other information about A? As stated, the difference that you want to bound can be made arbitrarily small.

Four distinct points in $\mathbb{R}^3$ with no three coplanar? I must be missing something.

Niles: no one has done this yet, at least no one that I know of.

Dear Pietro, Thank you for the answer. I understand that $M_f$ will not be right continuous in general. The question asks for what functions $f$ will $M_f$ be continuous?

Rbega, I noticed that I'd messed up critical points and values, I think it is fixed now.

Igor, while I am sure many of us are getting fed up with the rampant speculatory questions regarding Mochizuki and ABC, maybe the people asking are not regular enough on MO (or meta-MO) to be familiar with the ongoing debates about acceptability of such questions. This is not a crime. My impression is that this question is independent of Mochizuki's work and simply asks for why ABC implies FLT. Since this information is easily available on the internet, the question should be closed for not being research level rather than in annoyance at other recent ABC questions.

Gerhard: Valuation has at least three different meanings (at least one each in logic, measure theory and algebra according to wikipedia), none of which appear to be what the OP wants. In light of your comment, I'd recommend "weighted relation". For bonus points, a map $A \times B \to W$ for any $W$, ordered or otherwise, could be called a $W$-weighted relation.