  We give a brief exposition of the formulation of the bound state problem for
the one-dimensional system of $N$ attractive Dirac delta potentials, as an $N
\times N$ matrix eigenvalue problem ($\Phi A =\omega A$). The main aim of this
paper is to illustrate that the non-degeneracy theorem in one dimension breaks
down for the equidistantly distributed Dirac delta potential, where the matrix
$\Phi$ becomes a special form of the circulant matrix. We then give an
elementary proof that the ground state is always non-degenerate and the
associated wave function may be chosen to be positive by using the
Perron-Frobenius theorem. We also prove that removing a single center from the
system of $N$ delta centers shifts all the bound state energy levels upward as
a simple consequence of the Cauchy interlacing theorem.
