23,901 research outputs found
On invariant Schreier structures
Schreier graphs, which possess both a graph structure and a Schreier
structure (an edge-labeling by the generators of a group), are objects of
fundamental importance in group theory and geometry. We study the Schreier
structures with which unlabeled graphs may be endowed, with emphasis on
structures which are invariant in some sense (e.g. conjugation-invariant, or
sofic). We give proofs of a number of "folklore" results, such as that every
regular graph of even degree admits a Schreier structure, and show that, under
mild assumptions, the space of invariant Schreier structures over a given
invariant graph structure is very large, in that it contains uncountably many
ergodic measures. Our work is directly connected to the theory of invariant
random subgroups, a field which has recently attracted a great deal of
attention.Comment: 16 pages, added references and figure, to appear in L'Enseignement
Mathematiqu
A deterministic walk on the randomly oriented Manhattan lattice
Consider a randomly-oriented two dimensional Manhattan lattice where each
horizontal line and each vertical line is assigned, once and for all, a random
direction by flipping independent and identically distributed coins. A
deterministic walk is then started at the origin and at each step moves
diagonally to the nearest vertex in the direction of the horizontal and
vertical lines of the present location. This definition can be generalized, in
a natural way, to larger dimensions, but we mainly focus on the two dimensional
case. In this context the process localizes on two vertices at all large times,
almost surely. We also provide estimates for the tail of the length of paths,
when the walk is defined on the two dimensional lattice. In particular, the
probability of the path to be larger than decays sub-exponentially in .
It is easy to show that higher dimensional paths may not localize on two
vertices but will still eventually become periodic, and are therefore bounded.Comment: 18 pages, 12 figure
Weighted Modulo Orientations of Graphs
This dissertation focuses on the subject of nowhere-zero flow problems on graphs. Tutte\u27s 5-Flow Conjecture (1954) states that every bridgeless graph admits a nowhere-zero 5-flow, and Tutte\u27s 3-Flow Conjecture (1972) states that every 4-edge-connected graph admits a nowhere-zero 3-flow. Extending Tutte\u27s flows conjectures, Jaeger\u27s Circular Flow Conjecture (1981) says every 4k-edge-connected graph admits a modulo (2k+1)-orientation, that is, an orientation such that the indegree is congruent to outdegree modulo (2k+1) at every vertex. Note that the k=1 case of Circular Flow Conjecture coincides with the 3-Flow Conjecture, and the case of k=2 implies the 5-Flow Conjecture. This work is devoted to providing some partial results on these problems.
In Chapter 2, we study the problem of modulo 5-orientation for given multigraphic degree sequences. We prove that a multigraphic degree sequence d=(d1,..., dn) has a realization G with a modulo 5-orientation if and only if di≤1,3 for each i. In addition, we show that every multigraphic sequence d=(d1,..., dn) with min{1≤i≤n}di≥9 has a 9-edge-connected realization that admits a modulo 5-orientation for every possible boundary function. Jaeger conjectured that every 9-edge-connected multigraph admits a modulo 5-orientation, whose truth would imply Tutte\u27s 5-Flow Conjecture. Consequently, this supports the conjecture of Jaeger.
In Chapter 3, we show that there are essentially finite many exceptions for graphs with bounded matching numbers not admitting any modulo (2k+1)-orientations for any positive integers t. We additionally characterize all infinite many graphs with bounded matching numbers but without a nowhere-zero 3-flow. This partially supports Jaeger\u27s Circular Flow Conjecture and Tutte\u27s 3-Flow Conjecture.
In 2018, Esperet, De Verclos, Le and Thomass introduced the problem of weighted modulo orientations of graphs and indicated that this problem is closely related to modulo orientations of graphs, including Tutte\u27s 3-Flow Conjecture. In Chapter 4 and Chapter 5, utilizing properties of additive bases and contractible configurations, we reduced the Esperet et al\u27s edge-connectivity lower bound for some (signed) graphs families including planar graphs, complete graphs, chordal graphs, series-parallel graphs and bipartite graphs, indicating that much lower edge-connectivity bound still guarantees the existence of such orientations for those graph families.
In Chapter 6, we show that the assertion of Jaeger\u27s Circular Flow Conjecture with k=2 holds asymptotically almost surely for random 9-regular graphs
A Solvable Model of Secondary Structure Formation in Random Hetero-Polymers
We propose and solve a simple model describing secondary structure formation
in random hetero-polymers. It describes monomers with a combination of
one-dimensional short-range interactions (representing steric forces and
hydrogen bonds) and infinite range interactions (representing polarity forces).
We solve our model using a combination of mean field and random field
techniques, leading to phase diagrams exhibiting second-order transitions
between folded, partially folded and unfolded states, including regions where
folding depends on initial conditions. Our theoretical results, which are in
excellent agreement with numerical simulations, lead to an appealing physical
picture of the folding process: the polarity forces drive the transition to a
collapsed state, the steric forces introduce monomer specificity, and the
hydrogen bonds stabilise the conformation by damping the frustration-induced
multiplicity of states.Comment: 24 pages, 14 figure
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