||Recently there is a lot of interest in the study of Sturm-Liouville problems on graphs,|
called quantum graphs. However the study on cyclic quantum graphs are scarce. In
this thesis, we shall rst consider a characteristic function approach to the spectral
analysis for the Schrodinger operator H acting on graphene-like graphs|in nite periodic
hexagonal graphs with 3 distinct adjacent edges and 3 distinct potentials de ned
on them. We apply the Floquet-Bloch theory to derive a Floquet equation with parameters
theta_1, theta_2, whose roots de ne all the spectral values of H. Then we show that the
spectrum of this operator is continuous. Our results generalize those of Kuchment-Post
and Korotyaev-Lobanov. Our method is also simpler and more direct.
Next we solve two Ambarzumyan problems, one for graphene and another for a cyclic
graph with two vertices and 3 edges. Finally we solve an Hochstadt-Lieberman type
inverse spectral problem for the same cyclic graph with two vertices and 3 edges.
Keywords : quantum graphs, graphene, spectrum, Ambarzumyan problem, inverse