10 research outputs found
Ramified rectilinear polygons: coordinatization by dendrons
Simple rectilinear polygons (i.e. rectilinear polygons without holes or
cutpoints) can be regarded as finite rectangular cell complexes coordinatized
by two finite dendrons. The intrinsic -metric is thus inherited from the
product of the two finite dendrons via an isometric embedding. The rectangular
cell complexes that share this same embedding property are called ramified
rectilinear polygons. The links of vertices in these cell complexes may be
arbitrary bipartite graphs, in contrast to simple rectilinear polygons where
the links of points are either 4-cycles or paths of length at most 3. Ramified
rectilinear polygons are particular instances of rectangular complexes obtained
from cube-free median graphs, or equivalently simply connected rectangular
complexes with triangle-free links. The underlying graphs of finite ramified
rectilinear polygons can be recognized among graphs in linear time by a
Lexicographic Breadth-First-Search. Whereas the symmetry of a simple
rectilinear polygon is very restricted (with automorphism group being a
subgroup of the dihedral group ), ramified rectilinear polygons are
universal: every finite group is the automorphism group of some ramified
rectilinear polygon.Comment: 27 pages, 6 figure
Distance Related Graph Invariants in Triangulations and Quadrangulations of the Sphere
The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. I provide asymptotic upper bounds and sharp lower bounds for the Wiener index of simple triangulations and quadrangulations with given connectivity. Additionally, I make conjectures for the extremal triangulations and quadrangulations which maximize the Wiener index based on computational evidence. If σ(v) denotes the arithmetic mean of the distances from v to all other vertices of G, then the remoteness and proximity of G are defined as the largest and smallest value of σ(v) over all vertices v of G, respectively. I give sharp upper bounds on the remoteness and asymptotic upper bounds on the proximity of simple triangulations and quadrangulations of given order and connectivity
Spin foam models with finite groups
Spin foam models, loop quantum gravity and group field theory are discussed
as quantum gravity candidate theories and usually involve a continuous Lie
group. We advocate here to consider quantum gravity inspired models with finite
groups, firstly as a test bed for the full theory and secondly as a class of
new lattice theories possibly featuring an analogue diffeomorphism symmetry. To
make these notes accessible to readers outside the quantum gravity community we
provide an introduction to some essential concepts in the loop quantum gravity,
spin foam and group field theory approach and point out the many connections to
lattice field theory and condensed matter systems.Comment: 47 pages, 6 figure
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum