  Linear quadratic graphon field games (LQ-GFGs) are defined to be LQ games
which involve a large number of agents that are weakly coupled via a weighted
undirected graph on which each node represents an agent. The links of the graph
correspond to couplings between the agents' dynamics, as well as between the
individual cost functions, which each agent attempts to minimize. We formulate
limit LQ-GFG problems based on the assumption that these graphs lie in a
sequence which converges to a limit graphon. First, under a finite-rank
assumption on the limit graphon, the existence and uniqueness of solutions to
the formulated limit LQ-GFG problem is established. Second, based upon the
solutions to the limit LQ-GFG problem, epsilon-Nash equilibria are constructed
for the corresponding game problems with a very large but finite number of
players. This result is then generalized to the case with random initial
conditions. It is to be noted that LQ-GFG problems are distinct from the class
of graphon mean field game (GMFG) problems where a population is hypothesized
to be associated with each node of the graph [Caines and Huang CDC 2018, 2019].
