  We propose theoretical approach based on combination of graph theory and
generalized Ising model (GIM), which enables systematic determination of
extremal structures for crystalline solids without any information about
interactions or constituent elements. The conventional approach to find such
set of structure typically employs configurational polyhedra (CP) on
configuration space based on GIM description, whose vertices can always be
candidates to exhibit maximum or minimum physical quantities. We demonstrate
that the present approach can construct extended CP whose vertices not only
include those found in conventional CP, but also include other topologically
and/or configurationally characteristic structures on the same dimensional
configuration space with the same set of figures composed of underlying lattice
points, which therefore has significant advantage over the conventional
approach.
