  In this article we construct zeta functions of quantum graphs using a contour
integral technique based on the argument principle. We start by considering the
special case of the star graph with Neumann matching conditions at the center
of the star. We then extend the technique to allow any matching conditions at
the center for which the Laplace operator is self-adjoint and finally obtain an
expression for the zeta function of any graph with general vertex matching
conditions. In the process it is convenient to work with new forms for the
secular equation of a quantum graph that extend the well known secular equation
of the Neumann star graph. In the second half of the article we apply the zeta
function to obtain new results for the spectral determinant, vacuum energy and
heat kernel coefficients of quantum graphs. These have all been topics of
current research in their own right and in each case this unified approach
significantly expands results in the literature.
