  This paper studies linear quadratic graphon mean field games (LQ-GMFGs) with
common noise, in which a large number of agents are coupled via a weighted
undirected graph. One special feature, compared with the well-studied graphon
mean field games, is that the states of agents are described by the dynamic
systems with the idiosyncratic noises and common noise. The limit LQ-GMFGs with
common noise are formulated based on the assumption that these graphs lie in a
sequence converging to a limit graphon. By applying the spectral decomposition
method, the existence of solution for the formulated limit LQ-GMFGs is derived.
Moreover, based on the adequate convergence assumptions, a set of
$\epsilon$-Nash equilibrium strategies for the finite large population problem
is constructed.
