  We study the question of a modification of the running gauge coupling of
Yang-Mills theories due to quantum gravitational effects in a compact large
extra dimensional brane world scenario with a low energy quantum gravity scale.
The ADD scenario is applied for a D=d+\delta dimensional space-time in which
gravitons freely propagate, whereas the non-abelian gauge fields are confined
to a d-dimensional brane. The extra dimensions are taken to be toroidal and the
transverse fluctuation modes (branons) of the brane are taken into account. On
this basis we have calculated the one-loop corrections due to virtual
Kaluza-Klein graviton and branon modes for the gluon two- and three-point
functions in an effective field theory treatment. Applying momentum cut-off
regularization we find that for a d=4 brane the leading gravitational
divergencies cancel irrespective of the number of extra dimensions \delta,
generalizing previous results in the absence of extra-dimensions. Hence, again
the Yang-Mills \beta-function receives no gravitational corrections at
one-loop. This is no longer true in a `universal' extra dimensional scenario
with a d>4 dimensional brane. Moreover, the subleading power-law gravitational
divergencies induce higher-dimensional counterterms, which we establish in our
scheme. Interestingly, for d=4 these gravitationally induced counterterms are
of the form recently considered in non-abelian Lee-Wick extensions of the
standard model -- now with a possible mass scale in the TeV range due to the
presence of large extra dimensions.
