  Non-Hermitian systems exhibit diverse graph patterns of energy spectra under
open boundary conditions. Here we present an algebraic framework to
comprehensively characterize the spectral geometry and graph topology of
non-Bloch bands. Using a locally defined potential function, we unravel the
spectral-collapse mechanism from Bloch to non-Bloch bands, delicately placing
the spectral graph at the troughs of the potential landscape. The potential
formalism deduces non-Bloch band theory and generates the density of states via
Poisson equation. We further investigate the Euler-graph topology by
classifying spectral vertices based on their multiplicities and projections
onto the generalized Brillouin zone. Through concrete models, we identify three
elementary graph-topology transitions (UVY, PT-like, and self-crossing),
accompanied by the emergence of singularities in the generalized Brillouin
zone. Lastly, we unveil how to generally account for isolated edge states
outside the spectral graph. Our work lays the cornerstone for exploring the
versatile spectral geometry and graph topology of non-Hermitian non-Bloch
bands.
