  Exceptional points (EPs) are special parameter values of a non-Hermitian
eigenvalue problem where eigenfunctions corresponding to a multiple eigenvalue
coalesce. In optics, EPs are associated with a number of counter-intuitive wave
phenomena, and have potential applications in lasing, sensing, mode conversion
and spontaneous emission processes. For open photonic structures, resonant
states are complex-frequency solutions of the Maxwell's equations with outgoing
radiation conditions. For open dielectric structures without material gain or
loss, the eigenvalue problem for resonant states can have EPs, since it is
non-Hermitian due to radiation losses. For applications in nanophotonics, it is
important to understand EPs for resonant states on small finite dielectric
structures consisting of conventional dielectric materials. To achieve this
objective, we study EPs of resonant states on finite sets of parallel
infinitely-long circular dielectric cylinders with subwavelength radii. For
systems with two, three and four cylinders, we develop an efficient numerical
method for computing EPs, present examples for second and third order EPs, and
highlight their topological features. Our work provides insight to
understanding EPs on more complicated photonic structures, and can be used as a
simple platform to explore applications of EPs.
