38,603 research outputs found
Ihara's zeta function for periodic graphs and its approximation in the amenable case
In this paper, we give a more direct proof of the results by Clair and
Mokhtari-Sharghi on the zeta functions of periodic graphs. In particular, using
appropriate operator-algebraic techniques, we establish a determinant formula
in this context and examine its consequences for the Ihara zeta function.
Moreover, we answer in the affirmative one of the questions raised by
Grigorchuk and Zuk. Accordingly, we show that the zeta function of a periodic
graph with an amenable group action is the limit of the zeta functions of a
suitable sequence of finite subgraphs.Comment: 21 pages, 4 figure
Sandpiles, spanning trees, and plane duality
Let G be a connected, loopless multigraph. The sandpile group of G is a
finite abelian group associated to G whose order is equal to the number of
spanning trees in G. Holroyd et al. used a dynamical process on graphs called
rotor-routing to define a simply transitive action of the sandpile group of G
on its set of spanning trees. Their definition depends on two pieces of
auxiliary data: a choice of a ribbon graph structure on G, and a choice of a
root vertex. Chan, Church, and Grochow showed that if G is a planar ribbon
graph, it has a canonical rotor-routing action associated to it, i.e., the
rotor-routing action is actually independent of the choice of root vertex.
It is well-known that the spanning trees of a planar graph G are in canonical
bijection with those of its planar dual G*, and furthermore that the sandpile
groups of G and G* are isomorphic. Thus, one can ask: are the two rotor-routing
actions, of the sandpile group of G on its spanning trees, and of the sandpile
group of G* on its spanning trees, compatible under plane duality? In this
paper, we give an affirmative answer to this question, which had been
conjectured by Baker.Comment: 13 pages, 9 figure
Semiregular automorphisms of vertex-transitive graphs of certain valencies
AbstractIt is shown that a vertex-transitive graph of valency p+1, p a prime, admitting a transitive action of a {2,p}-group, has a non-identity semiregular automorphism. As a consequence, it is proved that a quartic vertex-transitive graph has a non-identity semiregular automorphism, thus giving a partial affirmative answer to the conjecture that all vertex-transitive graphs have such an automorphism and, more generally, that all 2-closed transitive permutation groups contain such an element (see [D. Marušič, On vertex symmetric digraphs, Discrete Math. 36 (1981) 69–81; P.J. Cameron (Ed.), Problems from the Fifteenth British Combinatorial Conference, Discrete Math. 167/168 (1997) 605–615])
Multi-way expanders and imprimitive group actions on graphs
For n at least 2, the concept of n-way expanders was defined by various
researchers. Bigger n gives a weaker notion in general, and 2-way expanders
coincide with expanders in usual sense. Koji Fujiwara asked whether these
concepts are equivalent to that of ordinary expanders for all n for a sequence
of Cayley graphs. In this paper, we answer his question in the affirmative.
Furthermore, we obtain universal inequalities on multi-way isoperimetric
constants on any finite connected vertex-transitive graph, and show that gaps
between these constants imply the imprimitivity of the group action on the
graph.Comment: Accepted in Int. Math. Res. Notices. 18 pages, rearrange all of the
arguments in the proof of Main Theorem (Theorem A) in a much accessible way
(v4); 14 pages, appendix splitted into a forthcoming preprint (v3); 17 pages,
appendix on noncommutative L_p spaces added (v2); 12 pages, no figure
Group actions on 1-manifolds: a list of very concrete open questions
This text focuses on actions on 1-manifolds. We present a (non exhaustive)
list of very concrete open questions in the field, each of which is discussed
in some detail and complemented with a large list of references, so that a
clear panorama on the subject arises from the lecture.Comment: 21 pages, 2 figure
A convergent string method: Existence and approximation for the Hamiltonian boundary-value problem
This article studies the existence of long-time solutions to the Hamiltonian
boundary value problem, and their consistent numerical approximation. Such a
boundary value problem is, for example, common in Molecular Dynamics, where one
aims at finding a dynamic trajectory that joins a given initial state with a
final one, with the evolution being governed by classical (Hamiltonian)
dynamics. The setting considered here is sufficiently general so that long time
transition trajectories connecting two configurations can be included, provided
the total energy is chosen suitably. In particular, the formulation
presented here can be used to detect transition paths between two stable basins
and thus to prove the existence of long-time trajectories. The starting point
is the formulation of the equation of motion of classical mechanics in the
framework of Jacobi's principle; a curve shortening procedure inspired by
Birkhoff's method is then developed to find geodesic solutions. This approach
can be viewed as a string method
Rank gradient, cost of groups and the rank versus Heegaard genus problem
We study the growth of the rank of subgroups of finite index in residually
finite groups, by relating it to the notion of cost.
As a by-product, we show that the `Rank vs. Heegaard genus' conjecture on
hyperbolic 3-manifolds is incompatible with the `Fixed Price problem' in
topological dynamics
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