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A002854
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Number of unlabeled Euler graphs with n nodes; number of unlabeled two-graphs with n nodes; number of unlabeled switching classes of graphs with n nodes; number of switching classes of unlabeled signed complete graphs on n nodes; number of Seidel matrices of order n.
(Formerly M0846 N0321)
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25
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1, 1, 2, 3, 7, 16, 54, 243, 2038, 33120, 1182004, 87723296, 12886193064, 3633057074584, 1944000150734320, 1967881448329407496, 3768516017219786199856, 13670271807937483065795200, 94109042015724412679233018144, 1232069666043220685614640133362240
(list;
graph;
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listen;
history;
text;
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OFFSET
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1,3
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COMMENTS
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Also called Eulerian graphs of strength 1.
"Switching" a graph 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.
"Switching" a signed simple graph at a node negates the signs of all edges incident with that node.
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.
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REFERENCES
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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.
CRC Handbook of Combinatorial Designs, 1996, p. 687.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 114, Eq. (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.
J. J. Seidel, A survey of two-graphs, pp. 481-511 of Colloquio Internazionale sulle Teorie Combinatorie (Roma, 1973), Vol. I, Accademia Nazionale dei Lincei, Rome, 1976; also pp. 146-176 in Geometry and Combinatorics: Selected Works of J.J. Seidel, ed. D.G. Corneil and R. Mathon, Academic Press, Boston, 1991..
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).
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LINKS
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R. W. Robinson, Enumeration of Euler graphs, pp. 147-153 of F. Harary, editor, Proof Techniques in Graph Theory. Academic Press, NY, 1969. (Annotated scanned copy)
T. Zaslavsky, Signed graphs, Discrete Appl. Math. 4 (1982), 47-74.
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FORMULA
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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(2*k+1)). [Robinson's formula, from Mallows & Sloane, simplified.] - M. F. Hasler, Apr 15 2012; corrected by Sean A. Irvine, Nov 05 2014
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EXAMPLE
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The a(4) = 3 Euler graphs on four nodes are:
1) o o 2) o-o 3) o-o
o o |/ | |
o o o-o
(End)
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PROG
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(PARI) A002854(n)={ /* Robinson's formula, simplified */ local(s=vector(n)); my( S=0, M()=sum( j=2, n, s[j]*sum( i=1, j-1, s[i]*gcd(i, j))) + sum( i=1, n, i*binomial(s[i], 2)+(i\2-1)*s[i]) + !!vecextract(s, 4^round(n/2)\3), inc()=!forstep(i=n, 1, -1, s[i]<n\i && s[i]++ && return; s[i]=0), t); until(inc(), t=0; for( i=1, n, if( n < t+=i*s[i], until(i++>n, s[i]=n); next(2))); t==n && S+=2^M()/prod(i=1, n, i^s[i]*s[i]!)); S} \\ M. F. Hasler, Apr 09 2012, adapted for current PARI version on Apr 12, 2018
(Python)
from itertools import combinations
from math import prod, factorial, gcd
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A002854(n): return int(sum(Fraction(1<<sum(p[r]*p[s]*gcd(r, s) for r, s in combinations(p.keys(), 2))+sum(((q>>1)-1)*r+(q*r*(r-1)>>1) for q, r in p.items())+any(q&1 for q in p), prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # Chai Wah Wu, Jul 03 2024
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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Name edited (changed "2-graph" to "two-graph" to avoid confusion with other 2-graphs) and comments on Eulerian graphs by Thomas Zaslavsky, Nov 21 2013
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STATUS
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approved
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