  For each positive integer $n$, the Fibonacci-sum graph $G_n$ on vertices
$1,2,\ldots,n$ is defined by two vertices forming an edge if and only if they
sum to a Fibonacci number. It is known that each $G_n$ is bipartite, and all
Hamiltonian paths in each $G_n$ have been classified. In this paper, it is
shown that each $G_n$ has at most one non-trivial automorphism, which is given
explicitly. Other properties of $G_n$ are also found, including the degree
sequence, the treewidth, the nature of the bipartition, and that $G_n$ is
outerplanar.
