Asymptotic Properties of Some Minor-Closed Classes of Graphs

Let${\cal A}$be a minor-closed class of labelled graphs, and let${\cal G}_{n}$be a random graph sampled uniformly from the set ofn-vertex graphs of${\cal A}$. Whennis large, what is the probability that${\cal G}_{n}$is connected? How many components does it have? How large is its biggest component? Thanks to the work of McDiarmid and his collaborators, these questions are now solved when all excluded minors are 2-connected.Using exact enumeration, we study a collection of classes${\cal A}$excluding non-2-connected minors, and show that their asymptotic behaviour may be rather different from the 2-connected case. This behaviour largely depends on the nature of the dominant singularity of the generating functionC(z) that counts connected graphs of${\cal A}$. We classify our examples accordingly, thus taking a first step towards a classification of minor-closed classes of graphs. Furthermore, we investigate a parameter that has not received any attention in this context yet: the size of the root component. It follows non-Gaussian limit laws (Beta and Gamma), and clearly merits a systematic investigation.

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Asymptotic properties of some minor-closed classes of graphs (conference version)

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2327 ◽

2013 ◽

Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽

Author(s):

Mireille Bousquet-Mélou ◽

Kerstin Weller

Keyword(s):

Asymptotic Properties ◽

Systematic Investigation ◽

Limit Laws ◽

Closed Class ◽

Connected Case ◽

International Audience ◽

Non Gaussian ◽

A Minor ◽

Excluded Minors

International audience Let $\mathcal{A}$ be a minor-closed class of labelled graphs, and let $G_n$ be a random graph sampled uniformly from the set of n-vertex graphs of $\mathcal{A}$. When $n$ is large, what is the probability that $G_n$ is connected? How many components does it have? How large is its biggest component? Thanks to the work of McDiarmid and his collaborators, these questions are now solved when all excluded minors are 2-connected. Using exact enumeration, we study a collection of classes $\mathcal{A}$ excluding non-2-connected minors, and show that their asymptotic behaviour is sometimes rather different from the 2-connected case. This behaviour largely depends on the nature of the dominant singularity of the generating function $C(z)$ that counts connected graphs of $\mathcal{A}$. We classify our examples accordingly, thus taking a first step towards a classification of minor-closed classes of graphs. Furthermore, we investigate a parameter that has not received any attention in this context yet: the size of the root component. This follows non-gaussian limit laws (beta and gamma), and clearly deserves a systematic investigation.

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Classification of Classical Non-Gaussian Noises with Respect to Their Detrimental Effects on the Evolution of Entanglement Using a System of Three-Qubit as Probe

International Journal of Theoretical Physics ◽

10.1007/s10773-019-04300-7 ◽

2019 ◽

Vol 58 (12) ◽

pp. 4278-4292

Author(s):

Lionel Tenemeza Kenfack ◽

Martin Tchoffo ◽

Lukong Cornelius Fai

Keyword(s):

Non Gaussian

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Asymptotic stability of heteroclinic cycles in systems with symmetry. II

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500003693 ◽

2004 ◽

Vol 134 (6) ◽

pp. 1177-1197 ◽

Cited By ~ 59

Author(s):

Martin Krupa ◽

Ian Melbourne

Keyword(s):

Asymptotic Stability ◽

Transition Matrix ◽

Sufficient Conditions ◽

Systematic Investigation ◽

Necessary And Sufficient Conditions ◽

Sufficient Condition ◽

Heteroclinic Cycles ◽

Higher Dimensional ◽

Necessary And Sufficient

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

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An Extension of Matroid Rank Submodularity and the $Z$-Rayleigh Property

The Electronic Journal of Combinatorics ◽

10.37236/600 ◽

2011 ◽

Vol 18 (1) ◽

Author(s):

Arun P. Mani

Keyword(s):

Half Plane ◽

Closed Class ◽

Tutte Polynomials ◽

A Minor

We define an extension of matroid rank submodularity called $R$-submodularity, and introduce a minor-closed class of matroids called extended submodular matroids that are well-behaved with respect to $R$-submodularity. We apply $R$-submodularity to study a class of matroids with negatively correlated multivariate Tutte polynomials called the $Z$-Rayleigh matroids. First, we show that the class of extended submodular matroids are $Z$-Rayleigh. Second, we characterize a minor-minimal non-$Z$-Rayleigh matroid using its $R$-submodular properties. Lastly, we use $R$-submodularity to show that the Fano and non-Fano matroids (neither of which is extended submodular) are $Z$-Rayleigh, thus giving the first known examples of $Z$-Rayleigh matroids without the half-plane property.

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Formulae and Asymptotics for Coefficients of Algebraic Functions

Combinatorics Probability Computing ◽

10.1017/s0963548314000728 ◽

2014 ◽

Vol 24 (1) ◽

pp. 1-53 ◽

Cited By ~ 17

Author(s):

CYRIL BANDERIER ◽

MICHAEL DRMOTA

Keyword(s):

Generating Functions ◽

Entire Functions ◽

Limit Laws ◽

Context Free Grammar ◽

Algebraic Functions ◽

Strongly Connected ◽

Non Gaussian ◽

Positive Coefficients ◽

Dyadic Numbers ◽

Context Free

We study the coefficients of algebraic functions ∑n≥0fnzn. First, we recall the too-little-known fact that these coefficientsfnalways admit a closed form. Then we study their asymptotics, known to be of the typefn~CAnnα. When the function is a power series associated to a context-free grammar, we solve a folklore conjecture: the critical exponents α cannot be 1/3 or −5/2; they in fact belong to a proper subset of the dyadic numbers. We initiate the study of the set of possible values forA. We extend what Philippe Flajolet called the Drmota–Lalley–Woods theorem (which states that α=−3/2 when the dependency graph associated to the algebraic system defining the function is strongly connected). We fully characterize the possible singular behaviours in the non-strongly connected case. As a corollary, the generating functions of certain lattice paths and planar maps are not determined by a context-free grammar (i.e., their generating functions are not ℕ-algebraic). We give examples of Gaussian limit laws (beyond the case of the Drmota–Lalley–Woods theorem), and examples of non-Gaussian limit laws. We then extend our work to systems involving non-polynomial entire functions (non-strongly connected systems, fixed points of entire functions with positive coefficients). We give several closure properties for ℕ-algebraic functions. We end by discussing a few extensions of our results (infinite systems of equations, algorithmic aspects).

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Random Graphs from a Minor-Closed Class

Combinatorics Probability Computing ◽

10.1017/s0963548309009717 ◽

2009 ◽

Vol 18 (4) ◽

pp. 583-599 ◽

Cited By ~ 20

Author(s):

COLIN McDIARMID

Keyword(s):

Poisson Distribution ◽

Random Graph ◽

Generating Function ◽

Random Graphs ◽

Giant Component ◽

Closed Class ◽

Exponential Generating Function ◽

Excluded Minor ◽

A Minor ◽

Labelled Graphs

A minor-closed class of graphs is addable if each excluded minor is 2-connected. We see that such a classof labelled graphs has smooth growth; and, for the random graphRnsampled uniformly from then-vertex graphs in, the fragment not in the giant component asymptotically has a simple ‘Boltzmann Poisson distribution’. In particular, asn→ ∞ the probability thatRnis connected tends to 1/A(ρ), whereA(x) is the exponential generating function forand ρ is its radius of convergence.

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The optical identification of radio sources

Symposium - International Astronomical Union ◽

10.1017/s007418090005138x ◽

1959 ◽

Vol 9 ◽

pp. 507-513

Author(s):

D. W. Dewhirst

Keyword(s):

Radio Sources ◽

Astronomy Observatory ◽

National Geographic Society ◽

Optical Identification ◽

National Geographic ◽

Extensive Search ◽

Sky Survey ◽

A Minor ◽

Minor Character

Previous attempts to identify any large proportion of the discrete sources discovered at meter wavelengths have met with small success. In the investigation briefly reported here an extensive search has been made on the original plates of the 48-inch Palomar—National Geographic Society Sky Survey, using the available published radio data, but more especially the as yet unpublished results of a survey between +50 and −10 degrees declination that has been made with the interferometer of the Mullard Radio Astronomy Observatory, Cambridge. This radio survey (3C) has been carried out at 159.5 Mc/s using the aerial array of the 2C survey [1] in modified form. An account of the observation and reduction of this recent survey is given by other speakers in the Symposium. The area of sky covered by the 3C survey, and the criteria for the selection and classification of the sources, are likely to undergo small extensions and modifications before the final catalog is ready for publication, but these modifications will be of a minor character and will not alter the general conclusions of the present paper.

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Moment Analysis of Eeg Amplitude Histograms and Spectral Analysis: Relative Classification of Several Behavioral Tasks

Perceptual and Motor Skills ◽

10.2466/pms.1993.76.3.859 ◽

1993 ◽

Vol 76 (3) ◽

pp. 859-866 ◽

Cited By ~ 8

Author(s):

Gary C. Galbraith ◽

Eugene H. Wong

Keyword(s):

Mental Arithmetic ◽

Spectral Power ◽

Amplitude Distribution ◽

Eyes Closed ◽

Behavioral Tasks ◽

Cast Doubt ◽

Non Gaussian ◽

Beta Frequency ◽

Single Predictor

Previous studies indicate that EEG amplitude probability density functions are Gaussian (normal) during rest and non-Gaussian during performance of mental tasks. In the present study we compared measures of normality, including higher central moments (e.g., skewness, kurtosis) and relative spectral power, to classify data sampled from several different behavioral tasks (resting eyes closed and mental arithmetic). Analysis shows significant classification in 22 of 25 subjects, based upon a total of 46 EEG variables. However, only two of these variables involved Gaussian properties of the amplitude distribution. Relative spectral power, on the other hand, contributed 33 predictor variables in delta, theta, alpha, and beta frequency bands (alpha was the best single predictor). These results lend support to studies demonstrating the robustness of EEG relative spectra but cast doubt upon the utility of Gaussian patterns in EEG amplitude distributions as predictors of behavioral states.

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