  The method of regions, which provides a systematic approach for computing
Feynman integrals involving multiple kinematic scales, proposes that a Feynman
integral can be approximated and even reproduced by summing over integrals
expanded in certain regions. A modern perspective of the method of regions
considers any given Feynman integral as a specific Newton polytope, defined as
the convex hull of the points associated with Symanzik polynomials. The regions
then correspond one-to-one with the lower facets of this polytope.
  As Symanzik polynomials correspond to the spanning trees and spanning 2-trees
of the Feynman graph, a graph-theoretical study of these polynomials may allow
us to identify the complete set of regions for a given expansion. In this work,
our primary focus is on three specific expansions: the on-shell expansion of
generic wide-angle scattering, the soft expansion of generic wide-angle
scattering, and the mass expansion of heavy-to-light decay. For each of these
expansions, we employ graph-theoretical approaches to derive the generic forms
of the regions involved in the method of regions. The results, applicable to
all orders, offer insights that can be leveraged to investigate various aspects
of scattering amplitudes.
