2,109 research outputs found
On the editing distance of graphs
An edge-operation on a graph is defined to be either the deletion of an
existing edge or the addition of a nonexisting edge. Given a family of graphs
, the editing distance from to is the smallest
number of edge-operations needed to modify into a graph from .
In this paper, we fix a graph and consider , the set of
all graphs on vertices that have no induced copy of . We provide bounds
for the maximum over all -vertex graphs of the editing distance from
to , using an invariant we call the {\it binary chromatic
number} of the graph . We give asymptotically tight bounds for that distance
when is self-complementary and exact results for several small graphs
Some colouring problems for Paley graphs
The Paley graph Pq, where q≡1(mod4) is a prime power, is the graph with vertices the elements of the finite field Fq and an edge between x and y if and only if x-y is a non-zero square in Fq. This paper gives new results on some colouring problems for Paley graphs and related discussion. © 2005 Elsevier B.V. All rights reserved
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Observations on the Lov\'asz -Function, Graph Capacity, Eigenvalues, and Strong Products
This paper provides new observations on the Lov\'{a}sz -function of
graphs. These include a simple closed-form expression of that function for all
strongly regular graphs, together with upper and lower bounds on that function
for all regular graphs. These bounds are expressed in terms of the
second-largest and smallest eigenvalues of the adjacency matrix of the regular
graph, together with sufficient conditions for equalities (the upper bound is
due to Lov\'{a}sz, followed by a new sufficient condition for its tightness).
These results are shown to be useful in many ways, leading to the determination
of the exact value of the Shannon capacity of various graphs, eigenvalue
inequalities, and bounds on the clique and chromatic numbers of graphs. Since
the Lov\'{a}sz -function factorizes for the strong product of graphs,
the results are also particularly useful for parameters of strong products or
strong powers of graphs. Bounds on the smallest and second-largest eigenvalues
of strong products of regular graphs are consequently derived, expressed as
functions of the Lov\'{a}sz -function (or the smallest eigenvalue) of
each factor. The resulting lower bound on the second-largest eigenvalue of a
-fold strong power of a regular graph is compared to the Alon--Boppana
bound; under a certain condition, the new bound is superior in its exponential
growth rate (in ). Lower bounds on the chromatic number of strong products
of graphs are expressed in terms of the order and the Lov\'{a}sz
-function of each factor. The utility of these bounds is exemplified,
leading in some cases to an exact determination of the chromatic numbers of
strong products or strong powers of graphs. The present research paper is aimed
to have tutorial value as well.Comment: electronic links to references were added in version 2; Available at
https://www.mdpi.com/1099-4300/25/1/10
Focal plane wavefront sensor achromatization : The multireference self-coherent camera
High contrast imaging and spectroscopy provide unique constraints for
exoplanet formation models as well as for planetary atmosphere models. But this
can be challenging because of the planet-to-star small angular separation and
high flux ratio. Recently, optimized instruments like SPHERE and GPI were
installed on 8m-class telescopes. These will probe young gazeous exoplanets at
large separations (~1au) but, because of uncalibrated aberrations that induce
speckles in the coronagraphic images, they are not able to detect older and
fainter planets. There are always aberrations that are slowly evolving in time.
They create quasi-static speckles that cannot be calibrated a posteriori with
sufficient accuracy. An active correction of these speckles is thus needed to
reach very high contrast levels (>1e7). This requires a focal plane wavefront
sensor. Our team proposed the SCC, the performance of which was demonstrated in
the laboratory. As for all focal plane wavefront sensors, these are sensitive
to chromatism and we propose an upgrade that mitigates the chromatism effects.
First, we recall the principle of the SCC and we explain its limitations in
polychromatic light. Then, we present and numerically study two upgrades to
mitigate chromatism effects: the optical path difference method and the
multireference self-coherent camera. Finally, we present laboratory tests of
the latter solution.
We demonstrate in the laboratory that the MRSCC camera can be used as a focal
plane wavefront sensor in polychromatic light using an 80 nm bandwidth at 640
nm. We reach a performance that is close to the chromatic limitations of our
bench: contrast of 4.5e-8 between 5 and 17 lambda/D.
The performance of the MRSCC is promising for future high-contrast imaging
instruments that aim to actively minimize the speckle intensity so as to detect
and spectrally characterize faint old or light gaseous planets.Comment: 14 pages, 20 figure
Combinatorics and geometry of finite and infinite squaregraphs
Squaregraphs were originally defined as finite plane graphs in which all
inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e.,
the vertices not incident with the outer face) have degrees larger than three.
The planar dual of a finite squaregraph is determined by a triangle-free chord
diagram of the unit disk, which could alternatively be viewed as a
triangle-free line arrangement in the hyperbolic plane. This representation
carries over to infinite plane graphs with finite vertex degrees in which the
balls are finite squaregraphs. Algebraically, finite squaregraphs are median
graphs for which the duals are finite circular split systems. Hence
squaregraphs are at the crosspoint of two dualities, an algebraic and a
geometric one, and thus lend themselves to several combinatorial
interpretations and structural characterizations. With these and the
5-colorability theorem for circle graphs at hand, we prove that every
squaregraph can be isometrically embedded into the Cartesian product of five
trees. This embedding result can also be extended to the infinite case without
reference to an embedding in the plane and without any cardinality restriction
when formulated for median graphs free of cubes and further finite
obstructions. Further, we exhibit a class of squaregraphs that can be embedded
into the product of three trees and we characterize those squaregraphs that are
embeddable into the product of just two trees. Finally, finite squaregraphs
enjoy a number of algorithmic features that do not extend to arbitrary median
graphs. For instance, we show that median-generating sets of finite
squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the
corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure
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