9,614 research outputs found
Regular Embeddings of Canonical Double Coverings of Graphs
AbstractThis paper addresses the question of determining, for a given graphG, all regular maps havingGas their underlying graph, i.e., all embeddings ofGin closed surfaces exhibiting the highest possible symmetry. We show that ifGsatisfies certain natural conditions, then all orientable regular embeddings of its canonical double covering, isomorphic to the tensor productGâK2, can be described in terms of regular embeddings ofG. This allows us to âliftâ the classification of regular embeddings of a given graph to a similar classification for its canonical double covering and to establish various properties of the âderivedâ maps by employing those of the âbaseâ maps. We apply these results to determining all orientable regular embeddings of the tensor productsKnâK2(known as the cocktail-party graphs) and of then-dipolesDn, the graphs consisting of two vertices and n parallel edges joining them. In the first case we show, in particular, that regular embeddings ofKnâK2exist only ifnis a prime powerpl, and there are 2Ï(nâ1) orÏ(nâ1) isomorphism classes of such maps (whereÏis Euler's function) according to whetherlis even or odd. Forleven an interesting new infinite family of regular maps is discovered. In the second case, orientable regular embeddings ofDnexist for each positive integern, and their number is a power of 2 depending on the decomposition ofninto primes
Quantum Gravity and Matter: Counting Graphs on Causal Dynamical Triangulations
An outstanding challenge for models of non-perturbative quantum gravity is
the consistent formulation and quantitative evaluation of physical phenomena in
a regime where geometry and matter are strongly coupled. After developing
appropriate technical tools, one is interested in measuring and classifying how
the quantum fluctuations of geometry alter the behaviour of matter, compared
with that on a fixed background geometry.
In the simplified context of two dimensions, we show how a method invented to
analyze the critical behaviour of spin systems on flat lattices can be adapted
to the fluctuating ensemble of curved spacetimes underlying the Causal
Dynamical Triangulations (CDT) approach to quantum gravity. We develop a
systematic counting of embedded graphs to evaluate the thermodynamic functions
of the gravity-matter models in a high- and low-temperature expansion. For the
case of the Ising model, we compute the series expansions for the magnetic
susceptibility on CDT lattices and their duals up to orders 6 and 12, and
analyze them by ratio method, Dlog Pad\'e and differential approximants. Apart
from providing evidence for a simplification of the model's analytic structure
due to the dynamical nature of the geometry, the technique introduced can shed
further light on criteria \`a la Harris and Luck for the influence of random
geometry on the critical properties of matter systems.Comment: 40 pages, 15 figures, 13 table
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