148 research outputs found
Pegging Graphs Yields a Small Diameter
We consider the following process for generating large random cubic graphs. Starting with a given graph, repeatedly add edges that join the midpoints of two randomly chosen edges. We show that the growing graph asymptotically almost surely has logarithmic diameter. This process is motivated by a particular type of peer-to-peer network. Our method extends to similar processes that generate regular graphs of higher degre
Cup Stacking in Graphs
Here we introduce a new game on graphs, called cup stacking, following a line
of what can be considered as -, -, or -person games such as chip
firing, percolation, graph burning, zero forcing, cops and robbers, graph
pebbling, and graph pegging, among others. It can be more general, but the most
basic scenario begins with a single cup on each vertex of a graph. For a vertex
with cups on it we can move all its cups to a vertex at distance from
it, provided the second vertex already has at least one cup on it. The object
is to stack all cups onto some pre-described target vertex. We say that a graph
is stackable if this can be accomplished for all possible target vertices.
In this paper we study cup stacking on many families of graphs, developing a
characterization of stackability in graphs and using it to prove the
stackability of complete graphs, paths, cycles, grids, the Petersen graph, many
Kneser graphs, some trees, cubes of dimension up to 20, "somewhat balanced"
complete -partite graphs, and Hamiltonian diameter two graphs. Additionally
we use the Gallai-Edmonds Structure Theorem, the Edmonds Blossom Algorithm, and
the Hungarian algorithm to devise a polynomial algorithm to decide if a
diameter two graph is stackable.
Our proof that cubes up to dimension 20 are stackable uses Kleitman's
Symmetric Chain Decomposition and the new result of Merino, M\"utze, and
Namrata that all generalized Johnson graphs (excluding the Petersen graph) are
Hamiltonian. We conjecture that all cubes and higher-dimensional grids are
stackable, and leave the reader with several open problems, questions, and
generalizations
Generation and properties of random graphs and analysis of randomized algorithms
We study a new method of generating random -regular graphs by
repeatedly applying an operation called pegging. The pegging
algorithm, which applies the pegging operation in each step, is a
method of generating large random regular graphs beginning with
small ones. We prove that the limiting joint distribution of the
numbers of short cycles in the resulting graph is independent
Poisson. We use the coupling method to bound the total variation
distance between the joint distribution of short cycle counts and
its limit and thereby show that is an upper bound
of the \eps-mixing time. The coupling involves two different,
though quite similar, Markov chains that are not time-homogeneous.
We also show that the -mixing time is not
. This demonstrates that the upper bound
is essentially tight. We study also the
connectivity of random -regular graphs generated by the pegging
algorithm. We show that these graphs are asymptotically almost
surely -connected for any even constant .
The problem of orientation of random hypergraphs is motivated by the
classical load balancing problem. Let be two fixed integers.
Let \orH be a hypergraph whose hyperedges are uniformly of size
.
To {\em -orient} a hyperedge, we assign exactly of its
vertices positive signs with respect to this hyperedge, and the rest
negative. A -orientation of \orH consists of a
-orientation of all hyperedges of \orH, such that each vertex
receives at most positive signs from its incident hyperedges.
When is large enough, we determine the threshold of the
existence of a -orientation of a random hypergraph. The
-orientation of hypergraphs is strongly related to a general
version of the off-line load balancing problem.
The other topic we discuss is computing the probability of induced
subgraphs in a random regular graph. Let and be a graph
on vertices. For any with , we compute the
probability that the subgraph of induced by
is . The result holds for any and is further
extended to , the probability space of
random graphs with given degree sequence . This result
provides a basic tool for studying properties, for instance the
existence or the counts, of certain types of induced subgraphs
Extremal Results for Peg Solitaire on Graphs
In a 2011 paper by Beeler and Hoilman, the game of peg solitaire is generalized to arbitrary boards. These boards are treated as graphs in the combinatorial sense. An open problem from that paper is to determine the minimum number of edges necessary for a graph with a fixed number of vertices to be solvable. This thesis provides new bounds on this number. It also provides necessary and sufficient conditions for two families of graphs to be solvable, along with criticality results, and the maximum number of pegs that can be left in each of the two graph families
Study of the cycle-to-cycle variations of an internal combustion engine fuelled with natural gas/hydrogen blends from the diagnosis of combustion pressure
Producción CientÃficaSe presenta una metodologÃa para el estudio de la influencia de la adicción de
hidrógeno (de 0 a 100% en sustitución del gas natural en mezclas de gas
natural/aire en la dispersión cÃclica de la combustión de un motor de encendido
provocado. Se utiliza un algoritmo genético que optimiza un modelo de
diagnóstico de la combustión a partir de la presión medida experimentalmente
en la cámara de combustión de forma que se optimizan entradas al modelo de
diágnóstico como son el ángulo de la presión máxima y el offset de presión, la
relación de compresión volumétrica, el Ãndice de la transferencia de calor a las
paredes, etc. Se observa que la adición de hidrógeno aumenta mucho la
velocidad de combustión a partir de un valor del 60%.Ministerio de economÃa y competitividad de Españ
Macroeconomic stabilisation and intervention policy under an exchange rate band
Exchange Rate;Stabilization;Foreign Exchange Market
An optical study of the structure of diffusion flame
Imperial Users onl
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