  Loop quantum gravity is based on a classical formulation of 3+1 gravity in
terms of a real SU(2) connection. Linearization of this classical formulation
about a flat background yields a description of linearised gravity in terms of
a {\em real} $U(1)\times U(1)\times U(1)$ connection. A `loop' representation,
in which holonomies of this connection are unitary operators, can be
constructed. These holonomies are not well defined operators in the standard
graviton Fock representation. We generalise our recent work on photons and U(1)
holonomies to show that Fock space gravitons are associated with distributional
states in the $U(1)\times U(1)\times U(1)$ loop representation. Our results may
illuminate certain aspects of the much deeper (and as yet unkown,) relation
between gravitons and states in nonperturbative loop quantum gravity.
  This work leans heavily on earlier seminal work by Ashtekar, Rovelli and
Smolin (ARS) on the loop representation of linearised gravity using {\em
complex} connections. In the last part of this work, we show that the loop
representation based on the {\em real} $U(1)\times U(1)\times U(1)$ connection
also provides a useful kinematic arena in which it is possible to express the
ARS complex connection- based results in the mathematically precise language
currently used in the field.
