  Polarization singularities including bound states in the continuum (BICs) and
circularly polarized states have provided promising opportunities in the
manipulation of light waves. Previous studies show that BICs in photonic
crystal slabs are protected by $C_{2}T$ symmetry and hence normally exist on
the high-symmetry lines of momentum space. Here, we propose an approach based
on graph theory to study these polarization singularities in momentum space,
especially in the region off the high-symmetry lines. With a polarization
graph, it is demonstrated for the first time that BICs can stably exist off the
high-symmetry lines of momentum space for both one-dimensional and
two-dimensional photonic crystal slabs. Furthermore, two kinds of interesting
processes, including the merging involved with this newly found BICs both on
and off the high-symmetry lines, are observed by merely changing the thickness
of photonic crystal slab. Our findings provide a new perspective to explore
polarization singularities in momentum space and render their further
applications in light-matter interaction and light manipulation.
