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A002854 Number of Euler graphs with n nodes; number of two-graphs with n nodes; number of switching classes of graphs with n nodes; number of Seidel matrices of order n.
(Formerly M0846 N0321)
1, 1, 2, 3, 7, 16, 54, 243, 2038, 33120, 1182004, 87723296, 12886193064, 3633057074584, 1944000150734320, 1967881448329407496, 3768516017219786199856, 13670271807937483065795200 (list; graph; refs; listen; history; text; internal format)



Also called Eulerian graphs of strength 1.

"Switching" at a node complements all the edges incident with that node. The illustration (see link) shows the 3 switching classes on 4 nodes. Switching at any node is the equivalence relation.

A graph is an Euler graph iff every node has even degree. It need not be connected. (Note that some graph theorists require an Euler graph to be connected so it has an Euler circuit, and call these graphs "even" graphs.)

The objects being counted in this sequence are unlabeled.


F. Buekenhout, ed., Handbook of Incidence Geometry, 1995, p. 881.

F. C. Bussemaker, R. A. Mathon and J. J. Seidel, Tables of two-graphs, T.H.-Report 79-WSK-05, Technological University Eindhoven, Dept. Mathematics, 1979; also pp. 71-112 of "Combinatorics and Graph Theory (Calcutta, 1980)", Lect. Notes Math. 885, 1981.

P. J. Cameron, Cohomological aspects of two-graphs, Math. Zeit., 157 (1977), 101-119.

CRC Handbook of Combinatorial Designs, 1996, p. 687.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 114, (4.7.1).

R. W. Robinson, Enumeration of Euler graphs, pp. 147-153 of F. Harary, editor, Proof Techniques in Graph Theory. Academic Press, NY, 1969.

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1979.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


Max Alekseyev, Table of n, a(n) for n = 1..60

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

P. J. Cameron and C. R. Johnson, The number of equivalence patterns of symmetric sign patterns, Discr. Math., 306 (2006), 3074-3077.

G. Greaves, J. H. Koolen, A. Munemasa, F. Szöllősi, Equiangular lines in Euclidean spaces, arXiv:1403.2155 [math.CO], 2014.

T. R. Hoffman, J. P. Solazzo, Complex Two-Graphs via Equiangular Tight Frames, arXiv preprint arXiv:1408.0334 [math.CO], 2014.

Michael Hofmeister, Counting double covers of graphs, Journal of graph theory 12.3 (1988): 437-444. (Beware of a typo!)

V. A. Liskovec, Enumeration of Euler Graphs, (in Russian), Akademiia Navuk BSSR, Minsk., 6 (1970), 38-46. (annotated scanned copy)

C. L. Mallows and N. J. A. Sloane, Two-graphs, switching classes and Euler graphs are equal in number, SIAM J. Appl. Math., 28 (1975), 876-880. (copy at N. J. A. Sloane's home page)

B. D. McKay, Eulerian graphs

R. E. Peile, Letter to N. J. A. Sloane, Feb 1989

R. C. Read, Letter to N. J. A. Sloane, Nov. 1976

N. J. A. Sloane, Switching classes of graphs with 4 nodes.

F. Szöllosi, P. R. J. Östergård, Enumeration of Seidel matrices, arXiv:1703.02943 [math.CO], 2017.

E. Weisstein's World of Mathematics, Eulerian Graph.


a(n) = sum_{s} 2^M(s)/product_{i} i^s(i)*s(i)!, where the sum is over n-tuples s in [0..n]^n such that n=sum i*s(i), M(s) = sum_{i<j} s(i)*s(j)*gcd(i,j) + sum_{i} (s(i)*(floor[i/2]-1)+i*binomial(s(i),2)) + sign(sum_{k} s(2k+1)). [Robinson's formula, from Mallows & Sloane, simplified.] - M. F. Hasler, Apr 15 2012; corrected by Sean A. Irvine, Nov 05 2014


From Joerg Arndt, Feb 05 2010: (Start)

The a(4) = 3 Euler graphs on four nodes are:


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(PARI) A002854(n)={ /* Robinson's formula, simplified */

my( s=0, N(S)=sum( j=2, #S, S[j]*sum( i=1, j-1, S[i]*gcd(i, j))) + sum( i=1, #S, i*binomial(S[i], 2)+(i\2-1)*S[i]) + !!vecextract(S, 4^round(#S/2)\3)); forvec( S=vector(n, i, [0, n\i]), my(t=0); for( i=1, n, if( n < t+=i*S[i], until(i++>n, S[i]=n); next(2))); t==n & s+=2^N(S)/prod(i=1, n, i^S[i]*S[i]!)); s} \\ - M. F. Hasler, Apr 09 2012


Cf. A003049, A085618, A085619, A085620, A007127, A133736.

Bisections: A182012, A182055.

Sequence in context: A089125 A289051 A282320 * A036356 A034732 A000278

Adjacent sequences:  A002851 A002852 A002853 * A002855 A002856 A002857




N. J. A. Sloane.


a(1)-a(26) are computed by R. W. Robinson.

More terms from Vladeta Jovovic, Apr 18 2000

Changed "2-graph" to "two-graph" in name (the usual name to avoid confusion with various "2-graphs").  Added alternative definition of "Euler graph" to clarify the issue. - Thomas Zaslavsky, Nov 21 2013



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Last modified November 19 01:44 EST 2017. Contains 294912 sequences.