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An asymptotic formula for the number of graphs having small diameter

Ioan Tomescu
1996 Discrete Mathematics  
In this paper it is shown that for every fixed k 1> 3, G(n; d = k) = 2(~) (6.2 -k + o(1))", where G(n; d = k) denotes the number of graphs of order n and diameter equal to k.  ...  It is also proved that for every fixed k>~2, lim,~G(n;d=k)/G(n;d=k+ 1)=lim.o~G(n;d=n-k)/ G(n;d=n-k+ 1)= oo hold.  ...  Acknowledgements I am indebted to one of the referees who pointed out an asymptotic formula for G(n; d = k) like to (5). Another referee indicated me that D.A.  ... 
doi:10.1016/0012-365x(94)00241-a fatcat:ejvbyvyda5hobolunclsdv6zsu

Page 1560 of Mathematical Reviews Vol. , Issue 2002C [page]

2002 Mathematical Reviews  
Summary: “In this paper some asymptotic formulas for the num- ber of graphs, digraphs or h-hypergraphs of order n and diameter equal to & are surveyed.  ...  2002c:05086 2002¢:05086 05C30 05C05 Chen, Xie Bin (PRC-ZZTC; Zhangzhou) An asymptotic enumeration theorem for the number of spanning trees in grids and tori. (Chinese.  ... 

A revised Moore bound for mixed graphs

Dominique Buset, Mourad El Amiri, Grahame Erskine, Mirka Miller, Hebert Pérez-Rosés
2016 Discrete Mathematics  
A 2008 paper of Nguyen and Miller derived an upper bound on the possible number of vertices of such graphs.  ...  The degree-diameter problem seeks to find the maximum possible order of a graph with a given (maximum) degree and diameter.  ...  this formula may have been extrapolated from the expressions for graphs of small diameter.  ... 
doi:10.1016/j.disc.2016.03.005 fatcat:ojh67ijjbbhs7prw4cdywynbii

Page 24 of Mathematical Reviews Vol. 49, Issue 1 [page]

1975 Mathematical Reviews  
In the present paper the author obtains an explicit formula and an asymptotic expression for H,(r) using the principle of inclusion and exclusion.  ...  For very small n, fewer edges may suffice, and it is not known what the smallest value of n, is for which the above formula holds whenever n = 7p.  ... 

The Evaluation of the Number and the Entropy of Spanning Trees on Generalized Small-World Networks

Raihana Mokhlissi, Dounia Lotfi, Joyati Debnath, Mohamed El Marraki, Noussaima EL Khattabi
2018 Journal of Applied Mathematics  
We propose a decomposition method for counting their number of spanning trees and we obtain the exact formulas, which are then verified by numerical simulations.  ...  Spanning trees have been widely investigated in many aspects of mathematics: theoretical computer science, combinatorics, so on. An important issue is to compute the number of these spanning trees.  ...  As an application of the number of spanning trees of a network, we use the entropy of spanning trees or what is called the asymptotic complexity (see, e.g., Dehmer, Emmert-Streib, Chen, Li, and Shi [2  ... 
doi:10.1155/2018/1017308 fatcat:xkdzdxchkbcbrpmtbqtcahujji

Random Networks Growing Under a Diameter Constraint [article]

Rajan M. Lukose, Lada A. Adamic
2003 arXiv   pre-print
The diameter constraint can be interpreted as an environmental selection pressure that may help explain the scale-free nature of graphs for which data is available at different times in their growth.  ...  We show that if the graph maintains the form of its degree distribution then that distribution must be approximately scale-free with an exponent between 2 and 3.  ...  The formula was derived by considering an ensemble of random graphs (without explicit clustering, see Refs.  ... 
arXiv:cond-mat/0301034v1 fatcat:lntgorjam5f7jj3h7fvfyahqj4

Epidemic Thresholds on Scale-Free Graphs: the Interplay between Exponent and Preferential Choice

Ph. Blanchard, C.-H. Chang, T. Krüger
2003 Annales de l'Institute Henri Poincare. Physique theorique  
We show for a model of scale-free graphs with biased partner choice that knowing the exponent for the degree distribution is in general not sufficient to decide epidemic threshold properties for exponents  ...  happens precisely when a positive fraction of the nodes form a cluster of bounded diameter.  ...  Of course the diameter of a random graph space is itself a random variable but it turns out that it has a small variation. For our purpose we therefore concentrate on the expected diameter.  ... 
doi:10.1007/s00023-003-0975-1 fatcat:vhrgoh5mene5jiqhvxcxov42da

Page 1583 of Mathematical Reviews Vol. 58, Issue 3 [page]

1979 Mathematical Reviews  
The author derives an asymptotic formula for the number of labelled graphs G, in the following seven classes of connected and strongly connected graphs: connected digraphs, anti-symmetric di- graphs, and  ...  Unfortunately, this simple implementation does not seem to produce an effective method for computing the desired numbers, except for graphs with very small numbers of points.  ... 

Page 1599 of Mathematical Reviews Vol. , Issue 2003C [page]

2003 Mathematical Reviews  
differentiating the equation log(}~ ant) = 2 nn. x ax): They also derive an explicit formula for a, using Cauchy’s Formula, treating e™?  ...  Liskovets (BE-AOS; Minsk) 2003¢:05115 05C30 05C20 Song, Junho; Lee, Changwoo An asymptotic formula for exp(;+~). (English summary) Commun. Korean Math. Soc. 17 (2002), no. 2, 363-370.  ... 

Results on Finite Wireless Sensor Networks: Connectivity and Coverage [article]

Ali Eslami, Mohammad Nekoui, and Hossein Pishro-Nik, F. Fekri
2012 arXiv   pre-print
Many analytic results for the connectivity, coverage, and capacity of wireless networks have been reported for the case where the number of nodes, n, tends to infinity (large-scale networks).  ...  The majority of these results have not been extended for small or moderate values of n; whereas in many practical networks, n is not very large.  ...  In fact, we saw an example of this phenomenon in the asymptotic formula for connectivity in (14). However, in small-scale networks boundary effects cannot be neglected.  ... 
arXiv:1211.2198v1 fatcat:yjzjgtbaxbfu3gguhed6lhgl54

Page 62 of Mathematical Reviews Vol. , Issue 2000a [page]

2000 Mathematical Reviews  
First, the asymptotic behavior of the number of trees with n vertices and diameter k = k(n), where (n—k)/n a as n— 0 for some constant 0 <a <1, is deter- mined.  ...  A graph is called AT-free if it does not have an AT. We show that there is an O(n‘) time algorithm to compute the maximum weight of an independent set for AT-free graphs.  ... 

Page 1860 of Mathematical Reviews Vol. , Issue 94d [page]

1994 Mathematical Reviews  
edge forwarding index problem for small graphs.  ...  In this paper, we derive asymptotic formulas for 7,(m,n) and P,(m,n) when m is fixed and derive the distribution for the root face valency.  ... 

The Fractal Dimension of SAT Formulas [chapter]

Carlos Ansótegui, Maria Luisa Bonet, Jesús Giráldez-Cru, Jordi Levy
2014 Lecture Notes in Computer Science  
We study the fractal dimension of SAT formulas, and show that most industrial families of formulas are self-similar, with a small fractal dimension.  ...  Modern SAT solvers have experienced a remarkable progress on solving industrial instances. Most of the techniques have been developed after an intensive experimental testing process.  ...  These edges have no effect for small values of r, but may drop down the number of tiles for big values of r. In some other cases (see hardware-bmc-ibm, for instance), there is a long tail.  ... 
doi:10.1007/978-3-319-08587-6_8 fatcat:buzy72ifgjdpnf5mna3qkcyedq

Comment on "Families and clustering in a natural numbers network"

Jeffrey D. Achter
2004 Physical Review E  
E 69 036106 (2004) ] constructs a family of graphs from subsets of the natural numbers, and numerically estimates diameter, degree and clustering.  ...  We give exact asymptotic formulas for these quantities, and thereby argue that number theory is a more appropriate tool than simulation.  ...  In [1], the author examines an infinite class of finite graphs constructed from the natural numbers.  ... 
doi:10.1103/physreve.70.058103 pmid:15600810 fatcat:dgfoov3dfnhdfe6zvihvprxpi4

Analysis of Wireless Ad-Hoc and Sensor Networks in Finite Regime

Hossein Pishro-Nik, Faramarz Fekri
2008 2008 5th Annual IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks  
In the past, many analytic results for wireless networks have been reported for the case where the number of nodes n in the network tends to infinity (large-scale networks).  ...  In this paper, we first show that previous asymptotic results provide poor approximations for the finite networks (small-scale networks).  ...  In Figure 2 , we compare the probability of having a disconnected graph for n = 100 and p = 1 derived by exhaustive simulations and the asymptotic result.  ... 
doi:10.1109/sahcn.2008.19 dblp:conf/secon/Pishro-NikF08 fatcat:nyzv2rl4tbfodbw4a664mxugeu
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