  Recently, M.\ Ab\'ert and T.\ Hubai studied the following problem. The
chromatic measure of a finite simple graph is defined to be the uniform
distribution on its chromatic roots. Ab\'ert and Hubai proved that for a
Benjamini-Schramm convergent sequence of finite graphs, the chromatic measures
converge in holomorphic moments. They also showed that the normalized log of
the chromatic polynomial converges to a harmonic real function outside a
bounded disc.
  In this paper we generalize their work to a wide class of graph polynomials,
namely, multiplicative graph polynomials of bounded exponential type. A special
case of our results is that for any fixed complex number $v_0$ the measures
arising from the Tutte polynomial $Z_{G_n}(z,v_0)$ converge in holomorphic
moments if the sequence $(G_n)$ of finite graphs is Benjamini--Schramm
convergent. This answers a question of Ab\'ert and Hubai in the affirmative.
Even in the original case of the chromatic polynomial, our proof is
considerably simpler.
