A049287 -id:A049287 - cp's OEIS frontend

A038784 An intermediate sequence for nonisomorphic circulant undirected p^2-graphs, indexed by odd primes p.

4, 9, 16, 64, 196, 1296, 3600, 35344, 1397124, 4804864, 213218404, 2754990144, 9976014400, 133023596176, 6663770519184, 342723467635264, 1281107924034624, 67756702226982976, 963748303995476224, 3643830108147610000, 198705767966065582336, 2876682497805092456704

Offset: 1

N. J. A. Sloane, May 04 2000

M. Klin, V. A. Liskovets and R. Poeschel, Analytical enumeration of circulant graphs with prime-squared vertices, Sem. Lotharingien de Combin., B36d, 1996, 36 pages.

Cf. A038781, A038783.

a(p^2) = A049287(p)^2.

More terms from Valery A. Liskovets, May 09 2001

More terms and offset corrected by Sean A. Irvine, Feb 14 2021

A070995 Number of nonisomorphic (undirected) Cayley graphs for the group Zp x Zp, where Zp is the elementary Abelian group of order p and p is prime. The sequence is index by primes, though starts with 1.

Original entry on oeis.org

1, 5, 50, 17794, 174685429024800, 1476099903835055889100, 569361345959217303084880851701375547158, 24894339520238610434672964029323166045198384692144, 221903632506534809770887023612289701531002339299063461384464526904412590996

Offset: 1

Marni Mishna, May 18 2002

nonn

The formula comes from a cycle index; There is a similar formula for directed Cayley graphs

C. Godsil, On Cayley graph isomorphisms, Ars, Combin., 15:231-246, 1983

B. Alspach and M. Mishna, Enumeration of Cayley graphs and digraphs, Discr. Math., 256 (2002), 527-539. [Note: In the third sum of Theorem 4.2 the exponent should be (p^2-1)/2/h(d). - _Sean A. Irvine_ and _Marni Mishna_, Jul 28 2024]
Sean A. Irvine, Java program (github)
M. Mishna, Cayley Graphs

Cf. A049287.

a(9) from Sean A. Irvine, Jul 22 2024

A344517 Minimum diameter of 4-regular circulant graphs of order n.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7

Offset: 4

Andres Cicuttin, May 21 2021

nonn

F. Boesch and Jhing-Fa Wang, Reliable circulant networks with minimum transmission delay, IEEE Transactions on Circuits and Systems, vol. 32, no. 12, pp. 1286-1291, December 1985, doi: 10.1109/TCS.1985.1085667.
Bevan, David et al. Large circulant graphs of fixed diameter and arbitrary degree. Ars Math. Contemp. 13 (2017): 275-291.

R. Feria-Puron, H. Perez-Roses, and J. Ryan, Searching for Large Circulant Graphs, arXiv:1503.07357 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Graph Diameter
Wikipedia, Circulant Graph

Cf. A049287, A075545, A285620.

mindiameter[n_]:=Module[{nmax,tab,stab},
nmax=Floor[n/2];
tab=Flatten[#,1]&@Table[Table[{n,i,j,GraphDiameter[CirculantGraph[n,{i,j}]]},{i,1,j-1}],{j,2,nmax}];
stab=Sort[tab,#1[[4]]<#2[[4]]&];
stab[[1]][[4]]//Return]
Table[mindiameter[n],{n,4,120}]
Table[Ceiling[(Sqrt[2n-1]-1)/2],{n,4,88}] (* Stefano Spezia, May 23 2021 *)

a(n) = ceiling((sqrt(2n-1)-1)/2).

A266479 Number of n-vertex simple graphs G_n for which n does not divide the number of labeled copies of G_n.

Original entry on oeis.org

0, 2, 2, 6, 3, 20, 4

Offset: 1

John P. McSorley, Dec 29 2015

Let G_n be an n-vertex simple graph, with a(G_n) automorphisms. Then l(G_n) = n!/a(G_n) is the number of labeled copies of G_n. So a(n) is the number of G_n for which n does not divide l(G_n).
For prime p, a(p) is the number of circulants of order p.
The number of circulants of order n is A049287(n).

			If n=3 then both G_3 = K_3 and its complement have a(G_3) = 6, so l(G_3) = 3!/6 = 1, and so 3 does not divide l(G_3); no other graphs G_3 satisfy this, so a(3)=2.

John P. McSorley, Smallest labelled class (and largest automorphism group) of a tree T_{s,t} and good labellings of a graph, preprint, (2016).
R. C. Read, R. J. Wilson, An Atlas of Graphs, Oxford Science Publications, Oxford University Press, (1998).
James Turner, Point-symmetric graphs with a prime number of points, Journal of Combinatorial Theory, vol. 3 (1967), 136-145.

A000088 minus A266478.

Cf. A049287.

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A038784 An intermediate sequence for nonisomorphic circulant undirected p^2-graphs, indexed by odd primes p.

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A070995 Number of nonisomorphic (undirected) Cayley graphs for the group Zp x Zp, where Zp is the elementary Abelian group of order p and p is prime. The sequence is index by primes, though starts with 1.

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A344517 Minimum diameter of 4-regular circulant graphs of order n.

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A266479 Number of n-vertex simple graphs G_n for which n does not divide the number of labeled copies of G_n.

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