### Abstract:

The focus of this thesis is the spectral structure of second order self-adjoint
differential operators on graphs.
Various function spaces on graphs are defined and we define, in terms of both
differential systems and the afore noted function spaces, boundary value problems
on graphs. A boundary value problem on a graph is shown to be spectrally
equivalent to a system with separated boundary conditions. An example
is provided to illustrate the fact that, for Sturm-Liouville operators on graphs,
self-adjointness does not necessarily imply regularity. We also show that since
the differential operators considered are self-adjoint the algebraic and geometric
eigenvalue multiplicities are equal. Asymptotic bounds for the eigenvalues
are found using matrix Pr¨ufer angle methods.
Techniques common in the area of elliptic partial differential equations are
used to give a variational formulation for boundary value problems on graphs.
This enables us to formulate an analogue of Dirichlet-Neumann bracketing
for boundary value problems on graphs as well as to establish a min-max
principle. This eigenvalue bracketing gives rise to eigenvalue asymptotics and
consequently eigenfunction asymptotics.
Asymptotic approximations to the Green’s functions of Sturm-Liouville boundary value problems on graphs are obtained. These approximations are used
to study the regularized trace of the differential operators associated with
these boundary value problems. Inverse spectral problems for Sturm-Liouville
boundary value problems on graphs resembling those considered in Halberg
and Kramer, A generalization of the trace concept, Duke Math. J. 27 (1960),
607-617, for Sturm-Liouville problems, and Pielichowski, An inverse spectral
problem for linear elliptic differential operators, Universitatis Iagellonicae Acta
Mathematica XXVII (1988), 239-246, for elliptic boundary value problems,
are solved.
Boundary estimates for solutions of non-homogeneous boundary value problems
on graphs are given. In particular, bounds for the norms of the boundary
values of solutions to the non-homogeneous boundary value problem in terms
of the norm of the non-homogeneity are obtained and the eigenparameter dependence
of these bounds is studied.
Inverse nodal problems on graphs are then considered. Eigenfunction and
eigenvalue asymptotic approximations are used to provide an asymptotic expression
for the spacing of nodal points on each edge of the graph from which
the uniqueness of the potential, for given nodal data, is deduced. An explicit
formula for the potential in terms of the nodal points and eigenvalues is given.