Wright and Stepanov-Wright coefficients

Voblyi, V. A.

doi:10.1007/bf01137454

Cited by 11 publications

(12 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…&1 by Bagaev and Dmitriev [3]; for various proofs see Vobly@$ [30], Bender et al [5,14,19]. Bolloba s [8] discovered that the leading term in the Wright's estimate holds essentially as an upper bound in a significantly wider range of l, namely…”

Section: And1mentioning

confidence: 97%

On the Largest Component of the Random Graph at a Nearcritical Stage

Pittel

2001

Journal of Combinatorial Theory, Series B

View full text Add to dashboard Cite

Section: And1mentioning

confidence: 97%

On the Largest Component of the Random Graph at a Nearcritical Stage

Pittel

2001

Journal of Combinatorial Theory, Series B

View full text Add to dashboard Cite

“…Stepanov ( 1987), for instance) that the total number of the subgraphs G of the complete graph Kn , which meet the conditions (1) and (2), is at most…”

Section: Jomentioning

confidence: 99%

The structure of a random graph at the point of the phase transition

Łuczak¹,

Pittel²,

Wierman³

1994

Trans. Amer. Math. Soc.

107

View full text Add to dashboard Cite

Abstract.Consider the random graph models G(n, # edges = M) and G{n, Prob(edge) = p) with M = M (ri) = (1 + Xn~ll3)n/2 and p = p(n) = (1 +Xn~ll*)/n . For / > -1 define an /-component of a random graph as a component which has exactly / more edges than vertices. Call an /-component with / > 1 a complex component. For both models, we show that when A is constant, the expected number of complex components is bounded, almost surely (a.s.) each of these components (if any exist) has size of order n2^ , and the maximum value of / is bounded in probability. We prove that a.s. the largest suspended tree in each complex component has size of order n2/3, and deletion of all suspended trees results in a "smoothed" graph of size of order n1'3, with the maximum vertex degree 3. The total number of branching vertices, i.e., of degree 3, is bounded in probability. Thus, each complex component is almost surely topologically equivalent to a 3-regular multigraph of a uniformly bounded size. Lengths of the shortest cycle and of the shortest path between two branching vertices of a smoothed graph are each of order n1/3 . We find a relatively simple integral formula for the limit distribution of the numbers of complex components, which implies, in particular, that all values of the "complexity spectrum" have positive limiting probabilities. We also answer questions raised by Erdös and Rényi back in 1960.

show abstract

“…An equivalent expansion with coeficients 3(k − 1)c k−1 , which equals Ω k /k! by (41), was given by Voblyȋ [55], see [24, (8.14) and (8.15) …”

Section: Airy Function Expansionsmentioning

confidence: 99%

Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas

Janson¹

2007

Probab. Surveys

140

189

View full text Add to dashboard Cite

This survey is a collection of various results and formulas by different authors on the areas (integrals) of five related processes, viz. Brownian motion, bridge, excursion, meander and double meander; for the Brownian motion and bridge, which take both positive and negative values, we consider both the integral of the absolute value and the integral of the positive (or negative) part. This gives us seven related positive random variables, for which we study, in particular, formulas for moments and Laplace transforms; we also give (in many cases) series representations and asymptotics for density functions and distribution functions. We further study Wright's constants arising in the asymptotic enumeration of connected graphs; these are known to be closely connected to the moments of the Brownian excursion area.The main purpose is to compare the results for these seven Brownian areas by stating the results in parallel forms; thus emphasizing both the similarities and the differences. A recurring theme is the Airy function which appears in slightly different ways in formulas for all seven random variables. We further want to give explicit relations between the many different similar notations and definitions that have been used by various authors. There are also some new results, mainly to fill in gaps left in the literature. Some short proofs are given, but most proofs are omitted and the reader is instead referred to the original sources.

show abstract

Wright and Stepanov-Wright coefficients

Cited by 11 publications

References 4 publications

On the Largest Component of the Random Graph at a Nearcritical Stage

On the Largest Component of the Random Graph at a Nearcritical Stage

The structure of a random graph at the point of the phase transition

Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas

Contact Info

Product

Resources

About