  D-brane probes, Hanany-Witten setups and geometrical engineering stand as a
trichotomy of the currently fashionable techniques of constructing gauge
theories from string theory. Meanwhile, asymptotic freedom, finitude and IR
freedom pose as a trichotomy of the beta-function behaviour in quantum field
theories. Parallel thereto is a trichotomy in set theory of finite, tame and
wild representation types. At the intersection of the above lies the theory of
quivers. We briefly review some of the terminology standard to the physics and
to the mathematics. Then we utilise certain results from graph theory and
axiomatic representation theory of path algebras to address physical issues
such as the implication of graph additivity to finiteness of gauge theories,
the impossibility of constructing completely IR free string orbifold theories
and the unclassifiability of N<2 Yang-Mills theories in four dimensions.
