  We study the effect of long-range connections on the infinite-randomness
fixed point associated with the quantum phase transitions in a transverse Ising
model (TIM). The TIM resides on a long-range connected lattice where any two
sites at a distance r are connected with a non-random ferromagnetic bond with a
probability that falls algebraically with the distance between the sites as
1/r^{d+\sigma}. The interplay of the fluctuations due to dilutions together
with the quantum fluctuations due to the transverse field leads to an
interesting critical behaviour. The exponents at the critical fixed point
(which is an infinite randomness fixed point (IRFP)) are related to the
classical "long-range" percolation exponents. The most interesting observation
is that the gap exponent \psi is exactly obtained for all values of \sigma and
d. Exponents depend on the range parameter \sigma and show a crossover to
short-range values when \sigma >= 2 -\eta_{SR} where \eta_{SR} is the anomalous
dimension for the conventional percolation problem. Long-range connections are
also found to tune the strength of the Griffiths phase.
