  Recently a type of robust exceptional points was found that is insensitive to
the coupling disorder in the bulk. Here we show that a disparity emerges when
the number of coupled cavities in this one-dimensional array changes from even
to odd. The robust exceptional point only exists in the former case, whereas
the location of the exceptional point in the latter depends inversely on the
size of the cavity array and is subjected to coupling disorder in the bulk. We
further show that the exceptional points in these two cases are second and
third order, respectively. We elucidate the origin of the robust EP as a
restricted bulk zero mode, which shares the same robustness against coupling
disorder and has a finite amplitude adjacent to the boundary. This finding
enables us to identify robust EPs in higher dimensional systems reliably, which
can exist in the presence of either sublattice symmetry or the non-Hermitian
particle-hole symmetry evolved from it.
