  We identify the poles and zeros of the scattering matrix of a simple quantum
graph by means of systematic measurement and analysis of Wigner, transmission,
and reflection complex time delays. We examine the ring graph because it
displays both shape and Feshbach resonances, the latter of which arises from an
embedded eigenstate on the real frequency axis. Our analysis provides a unified
understanding of the so-called shape, Feshbach, electromagnetically-induced
transparency, and Fano resonances, on the basis of the distribution of poles
and zeros of the scattering matrix in the complex frequency plane. It also
provides a first-principles understanding of sharp resonant scattering
features, and associated large time delay, in a variety of practical devices,
including photonic microring resonators, microwave ring resonators, and
mesoscopic ring-shaped conductor devices. Our analysis is the first use of
reflection time difference, as well as the first comprehensive use of complex
time delay, to analyze experimental scattering data.
