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Search: a033483 -id:a033483
Displaying 1-10 of 17 results found. page 1 2
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A006820 Number of connected regular simple graphs of degree 4 (or quartic graphs) with n nodes.
(Formerly M1617) 		+10
33
1, 0, 0, 0, 0, 1, 1, 2, 6, 16, 59, 265, 1544, 10778, 88168, 805491, 8037418, 86221634, 985870522, 11946487647, 152808063181, 2056692014474, 29051272833609, 429668180677439, 6640165204855036, 107026584471569605, 1796101588825595008, 31333997930603283531, 567437240683788292989 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,8
COMMENTS
The null graph on 0 vertices is vacuously connected and 4-regular. - Jason Kimberley, Jan 29 2011
The Multiset Transform of this sequence gives a triangle which gives in row n and column k the 4-regular simple graphs with n>=1 nodes and k>=1 components (row sums A033301), starting:
;
;
;
;
1 ;
1 ;
2 ;
6 ;
16 ;
59 1 ;
265 1 ;
1544 3 ;
10778 8 ;
88168 25 ;
805491 87 1 ;
8037418 377 1 ;
86221634 2023 3 ;
985870522 13342 9 ;
11946487647 104568 27 ;
152808063181 930489 96 1 ; - R. J. Mathar, Jun 02 2022
REFERENCES
CRC Handbook of Combinatorial Designs, 1996, p. 648.
I. A. Faradzev, Constructive enumeration of combinatorial objects, pp. 131-135 of Problèmes combinatoires et théorie des graphes (Orsay, 9-13 Juillet 1976). Colloq. Internat. du C.N.R.S., No. 260, Centre Nat. Recherche Scient., Paris, 1978.
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Table of n, a(n) for n=0..28.
Wayne Barrett, Shaun Fallat, Veronika Furst, Shahla Nasserasr, Brendan Rooney, and Michael Tait, Regular Graphs of Degree at most Four that Allow Two Distinct Eigenvalues, arXiv:2305.10562 [math.CO], 2023. See p. 7.
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146. [Jason Kimberley, Nov 24 2009]
M. Meringer, GenReg, Generation of regular graphs, program.
Markus Meringer, H. James Cleaves, Stephen J. Freeland, Beyond Terrestrial Biology: Charting the Chemical Universe of α-Amino Acid Structures, Journal of Chemical Information and Modeling, 53.11 (2013), pp. 2851-2862.
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Quartic Graph
Eric Weisstein's World of Mathematics, Regular Graph
Zhipeng Xu, Xiaolong Huang, Fabian Jimenez, and Yuefan Deng, A new record of enumeration of regular graphs by parallel processing, arXiv:1907.12455 [cs.DM], 2019.
FORMULA
a(n) = A184943(n) + A033886(n).
a(n) = A033301(n) - A033483(n).
Inverse Euler transform of A033301.
Row sums of A184940. - R. J. Mathar, May 30 2022
CROSSREFS
From Jason Kimberley, Mar 27 2010 and Jan 29 2011: (Start)
4-regular simple graphs: this sequence (connected), A033483 (disconnected), A033301 (not necessarily connected).
Connected regular simple graphs: A005177 (any degree), A068934 (triangular array); specified degree k: A002851 (k=3), this sequence (k=4), A006821 (k=5), A006822 (k=6), A014377 (k=7), A014378 (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).
Connected 4-regular simple graphs with girth at least g: this sequence (g=3), A033886 (g=4), A058343 (g=5), A058348 (g=6).
Connected 4-regular simple graphs with girth exactly g: A184943 (g=3), A184944 (g=4), A184945 (g=5).
Connected 4-regular graphs: this sequence (simple), A085549 (multigraphs with loops allowed), A129417 (multigraphs with loops verboten). (End)
KEYWORD
nonn,nice,hard
AUTHOR
N. J. A. Sloane
EXTENSIONS
a(19)-a(22) were appended by Jason Kimberley on Sep 04 2009, Nov 24 2009, Mar 27 2010, and Mar 18 2011, from running M. Meringer's GENREG for 3.4, 44, and 403 processor days, and 15.5 processor years, at U. Ncle.
a(22) corrected and a(23)-a(28) from Andrew Howroyd, Mar 10 2020
STATUS
approved
A165652 Number of disconnected 2-regular graphs on n vertices. +10
23
0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 8, 9, 12, 16, 20, 24, 32, 38, 48, 59, 72, 87, 109, 129, 157, 190, 229, 272, 330, 390, 467, 555, 659, 778, 926, 1086, 1283, 1509, 1774, 2074, 2437, 2841, 3322, 3871, 4509, 5236, 6094, 7055, 8181, 9464, 10944, 12624, 14577, 16778, 19322, 22209 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,9
COMMENTS
a(n) is also the number of partitions of n such that each part i satisfies 2<i<n.
For n>=2, it appears that a(n+1) is the number of (1,0)-separable partitions of n, as defined at A239482. For example, the four (1,0)-separable partitions of 9 are 621, 531, 441, 31212, corresponding to a(10) = 4. - Clark Kimberling, Mar 21 2014.
LINKS
Andrew van den Hoeven, Table of n, a(n) for n = 0..10000
Jason Kimberley, Disconnected regular graphs (with girth at least 3)
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g
FORMULA
a = A008483 - A179184 = Euler_tranformation(A179184) - A179184.
For n > 2, since there is exactly one connected 2-regular graph on n vertices (the n cycle C_n) then a(n) = A008483(n) - 1.
(A008483(n) is also the number of not necessarily connected 2-regular graphs on n vertices.)
Column D(n, 2) in the triangle A068933.
EXAMPLE
The a(6)=1 graph is C_3+C_3. The a(7)=1 graph is C_3+C_4. The a(8)=2 graphs are C_3+C_5, C_4+C_4. The a(9)=3 graphs are 3C_3, C_3+C_6, C_4+C_5.
PROG
(Magma) p := NumberOfPartitions; a := func< n | n lt 3 select 0 else p(n) - p(n-1) - p(n-2) + p(n-3) - 1 >;
CROSSREFS
2-regular simple graphs: A179184 (connected), this sequence (disconnected), A008483 (not necessarily connected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A157928 (k=0), A157928 (k=1), this sequence (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8).
Disconnected 2-regular simple graphs with girth at least g: this sequence (g=3), A185224 (g=4), A185225 (g=5), A185226 (g=6), A185227 (g=7), A185228 (g=8), A185229 (g=9).
Cf. A239482.
KEYWORD
easy,nonn
AUTHOR
Jason Kimberley, Sep 28 2009
STATUS
approved
A033301 Number of 4-valent (or quartic) graphs with n nodes. +10
17
1, 0, 0, 0, 0, 1, 1, 2, 6, 16, 60, 266, 1547, 10786, 88193, 805579, 8037796, 86223660, 985883873, 11946592242, 152808993767, 2056701139136, 29051369533596, 429669276147047, 6640178380127244, 107026751932268789, 1796103830404560857, 31334029441145918974, 567437704731717802783 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,8
COMMENTS
Because the triangle A051031 is symmetric, a(n) is also the number of (n-5)-regular graphs on n vertices. - Jason Kimberley, Sep 22 2009
REFERENCES
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
LINKS
Table of n, a(n) for n=0..28.
M. Meringer, Tables of Regular Graphs
M. Meringer, Erzeugung Regulaerer Graphen, Diploma thesis, University of Bayreuth, January 1996. [From Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010]
N. J. A. Sloane, Transforms
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Quartic Graph
FORMULA
Euler transform of A006820. - Martin Fuller, Dec 04 2006
MATHEMATICA
A006820 = Cases[Import["https://oeis.org/A006820/b006820.txt", "Table"], {_, _}][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A006820] (* Jean-François Alcover, Nov 26 2019, updated Mar 17 2020 *)
CROSSREFS
4-regular simple graphs: A006820 (connected), A033483 (disconnected), this sequence (not necessarily connected).
Regular graphs A005176 (any degree), A051031 (triangular array), chosen degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7).
KEYWORD
nonn,nice,hard
AUTHOR
Ronald C. Read
EXTENSIONS
a(16) from Axel Kohnert (kohnert(AT)uni-bayreuth.de), Jul 24 2003
a(17)-a(19) from Jason Kimberley, Sep 12 2009
a(20)-a(21) from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010
a(22) from Jason Kimberley, Oct 15 2011
a(22) corrected and a(23)-a(28) from Andrew Howroyd, Mar 08 2020
STATUS
approved
A185244 Number of disconnected 4-regular simple graphs on n vertices with girth at least 4. +10
17
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 15, 35, 247, 1692, 17409, 197924, 2492824, 33117880, 461597957, 6709514218, 101153412903, 1597440868898 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,19
LINKS
Table of n, a(n) for n=0..31.
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g
FORMULA
a(n) = A185344(n) - A033886(n) = Euler_transformation(A033886)(n) - A033886(n).
a(n) = A185044(n) + A185245(n).
CROSSREFS
4-regular simple graphs with girth at least 4: A033886 (connected), this sequence (disconnected), A185344 (not necessarily connected).
Disconnected 4-regular simple graphs with girth at least g: A033483 (g=3), this sequence (g=4), A185245 (g=5), A185246 (g=6).
Disconnected k-regular simple graphs with girth at least 4: A185214 (any k), A185204 (triangle); specified degree k: A185224 (k=2), A185234 (k=3), this sequence (k=4), A185254 (k=5), A185264 (k=6), A185274 (k=7), A185284 (k=8), A185294 (k=9).
KEYWORD
nonn,more,hard
AUTHOR
Jason Kimberley, Feb 22 2011
EXTENSIONS
a(31) appended by the author once A033886(23) was known, Nov 03 2011
a(31) corrected by the author, Jan 05 2013
STATUS
approved
A165653 Number of disconnected 3-regular (cubic) graphs on 2n vertices. +10
12
0, 0, 0, 0, 1, 2, 9, 31, 147, 809, 5855, 54477, 633057, 8724874, 137047391, 2391169355, 45626910415, 942659626031, 20937539944549, 497209670658529, 12566853576025106, 336749273734805530, 9534909974420181226 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
Table of n, a(n) for n=0..22.
Jason Kimberley, Disconnected regular graphs (with girth at least 3)
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g
Eric Weisstein's World of Mathematics, Cubic Graph
Eric Weisstein's World of Mathematics, Disconnected Graph
FORMULA
a(n) = A005638(n) - A002851(n).
a(n) = A068933(2n, 3).
MATHEMATICA
A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {_, _}][[All, 2]]];
A005638 = A@005638;
A002851 = A@002851;
a[n_] := A005638[[n + 1]] - A002851[[n + 1]];
a /@ Range[0, 20] (* Jean-François Alcover, Jan 21 2020 *)
CROSSREFS
3-regular simple graphs: A002851 (connected), this sequence (disconnected), A005638 (not necessarily connected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), this sequence (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8), A185293 (k=9), A185203 (k=10), A185213 (k=11).
KEYWORD
nonn,hard
AUTHOR
Jason Kimberley, Sep 28 2009
STATUS
approved
A165655 Number of disconnected 5-regular (quintic) graphs on 2n vertices. +10
12
0, 0, 0, 0, 0, 0, 1, 3, 66, 8029, 3484760, 2595985770, 2815099031417, 4230059694039460, 8529853839173455678, 22496718465713456081402, 75951258300080722467845995, 322269241532759484921710401976 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,8
LINKS
Table of n, a(n) for n=0..17.
N. J. A. Sloane, Transforms
Jason Kimberley, Disconnected regular graphs (with girth at least 3)
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g
Eric Weisstein's World of Mathematics, Disconnected Graph
Eric Weisstein's World of Mathematics, Quintic Graph
FORMULA
a = A165626 - A006821 = Euler_transformation(A006821) - A006821.
a(n)=A068933(2n,5).
CROSSREFS
5-regular simple graphs: A006821 (connected), this sequence (disconnected), A165626 (not necessarily connected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), A033483 (k=4), this sequence (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8), A185293 (k=9), A185203 (k=10), A185213 (k=11).
KEYWORD
nonn,hard,more
AUTHOR
Jason Kimberley, Sep 28 2009
EXTENSIONS
Terms a(13)-a(17), due to the extension of A006821 by Andrew Howroyd, from Jason Kimberley, Mar 12 2020
STATUS
approved
A165656 Number of disconnected 6-regular (sextic) graphs on n vertices. +10
12
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 25, 297, 8199, 377004, 22014143, 1493574756, 114880777582, 9919463450855, 955388277929620, 102101882472479938, 12050526046888229845, 1563967741064673811531, 222318116370232302781485, 34486536277291555593662301, 5817920265098158804699762770 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,17
LINKS
Table of n, a(n) for n=0..31.
N. J. A. Sloane, Transforms
Jason Kimberley, Disconnected regular graphs (with girth at least 3)
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g
Eric Weisstein's World of Mathematics, Disconnected Graph
Eric Weisstein's World of Mathematics, Regular Graph
Eric Weisstein's World of Mathematics, Sextic Graph
FORMULA
a = A165627 - A006822 = Euler_transformation(A006822) - A006822.
a(n) = D(n, 6) in the triangle A068933.
CROSSREFS
6-regular simple graphs: A006822 (connected), this sequence (disconnected), A165627 (not necessarily connected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), this sequence (k=6), A165877 (k=7), A165878 (k=8), A185293 (k=9), A185203 (k=10), A185213 (k=11).
KEYWORD
nonn,hard
AUTHOR
Jason Kimberley, Sep 28 2009
EXTENSIONS
Terms a(25) and beyond from Andrew Howroyd, May 20 2020
STATUS
approved
A165877 Number of disconnected 7-regular (septic) graphs on 2n vertices. +10
11
0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 1562, 21617036, 733460349818, 42703733735064572, 4073409466378991404239, 613990076321940092226829047, 141518698937232822678583027258225 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,10
LINKS
Table of n, a(n) for n=0..16.
N. J. A. Sloane, Transforms
Jason Kimberley, Disconnected regular graphs (with girth at least 3)
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g
Eric Weisstein's World of Mathematics, Septic Graph
FORMULA
a = A165628 - A014377 = Euler_transformation(A014377) - A014377.
a(n)=D(2n, 7) in the triangle A068933.
CROSSREFS
7-regular simple graphs: A014377 (connected), this sequence(disconnected), A165628 (not necessarily connected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), this sequence (k=7), A165878 (k=8), A185293 (k=9), A185203 (k=10), A185213 (k=11).
KEYWORD
nonn,hard,more
AUTHOR
Jason Kimberley, Sep 28 2009
EXTENSIONS
a(13)-a(16) from Andrew Howroyd, May 20 2020
STATUS
approved
A165878 Number of disconnected 8-regular simple graphs on n vertices. +10
11
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 7, 100, 10901, 3470736, 1473822243, 734843169811, 423929978716908, 281768931380519766, 215039290728074333738, 187766225244288486398132, 186874272297562916477691894, 211165081721567703008217979077 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,21
LINKS
Table of n, a(n) for n=0..31.
Jason Kimberley, Disconnected regular graphs (with girth at least 3)
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g
Eric Weisstein's World of Mathematics, Disconnected Graph
Eric Weisstein's World of Mathematics, Octic Graph
FORMULA
a = A180260 - A014378 = Euler_transformation(A014378) - A014378.
a(n) = D(n, 8) in the triangle A068933.
EXAMPLE
The a(18)=1 graph is K_9+K_9.
CROSSREFS
8-regular simple graphs: A014378 (connected), this sequence (disconnected), A180260 (not necessarily connected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), this sequence (k=8), A185293 (k=9), A185203 (k=10), A185213 (k=11).
KEYWORD
nonn,hard
AUTHOR
Jason Kimberley, Sep 29 2009
EXTENSIONS
Terms a(26) and beyond from Andrew Howroyd, May 20 2020
STATUS
approved
A185203 Number of disconnected 10-regular graphs with n nodes. +10
9
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 11, 550, 806174, 2585947720, 9802278927562, 42709859521915286, 214798119408798346811, 1251607430636395979871600, 8463468717232507491862780325, 66406919318277846825588474735084 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,25
LINKS
Table of n, a(n) for n=0..33.
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g
CROSSREFS
10-regular simple graphs: A014382 (connected), this sequence (disconnected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8), A185293 (k=9), this sequence (k=10), A185213 (k=11).
KEYWORD
nonn,hard
AUTHOR
Jason Kimberley, Jan 26 2012
EXTENSIONS
Terms a(29) and beyond from Andrew Howroyd, May 20 2020
STATUS
approved
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Last modified August 30 00:57 EDT 2024. Contains 375520 sequences. (Running on oeis4.)