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An asymptotic formula for the number of graphs having small diameter

1996
*
Discrete Mathematics
*

In this paper it is shown that

doi:10.1016/0012-365x(94)00241-a
fatcat:ejvbyvyda5hobolunclsdv6zsu
*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. ...##
###
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

2016
*
Discrete Mathematics
*

A 2008 paper

doi:10.1016/j.disc.2016.03.005
fatcat:ojh67ijjbbhs7prw4cdywynbii
*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*. ...##
###
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

2018
*
Journal of Applied Mathematics
*

We propose a decomposition method

doi:10.1155/2018/1017308
fatcat:xkdzdxchkbcbrpmtbqtcahujji
*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 ...##
###
Random Networks Growing Under a Diameter Constraint
[article]

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. ...

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

2003
*
Annales de l'Institute Henri Poincare. Physique theorique
*

We show

doi:10.1007/s00023-003-0975-1
fatcat:vhrgoh5mene5jiqhvxcxov42da
*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*. ...##
###
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]

2012
*
arXiv
*
pre-print

Many analytic results

arXiv:1211.2198v1
fatcat:yjzjgtbaxbfu3gguhed6lhgl54
*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. ...##
###
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]

2014
*
Lecture Notes in Computer Science
*

We study

doi:10.1007/978-3-319-08587-6_8
fatcat:buzy72ifgjdpnf5mna3qkcyedq
*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. ...##
###
Comment on "Families and clustering in a natural numbers network"

2004
*
Physical Review E
*

E 69 036106 (2004) ] constructs a family

doi:10.1103/physreve.70.058103
pmid:15600810
fatcat:dgfoov3dfnhdfe6zvihvprxpi4
*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*. ...##
###
Analysis of Wireless Ad-Hoc and Sensor Networks in Finite Regime

2008
*
2008 5th Annual IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks
*

In

doi:10.1109/sahcn.2008.19
dblp:conf/secon/Pishro-NikF08
fatcat:nyzv2rl4tbfodbw4a664mxugeu
*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. ...
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