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Enumeration of Cubic Graphs by Inclusion–Exclusion
Journal of Combinatorial Theory, Series A 86, 151157 (1999) Article ID jcta.1998.2923, available online at http:www.idealibrary.com on
NOTE Enumeration of Cubic Graphs by InclusionExclusion William Y. C. Chen Los Alamos National Laboratory, T-7, Mail Stop B284, Los Alamos, New Mexico 87545, and Center for Combinatorics, Nankai University, Tianjin 300071, People's Republic of China E-mail: chent7.lanl.gov, chenpublic.tpt.tj.cn
and James D. Louck Los Alamos National Laboratory, T-7, Mail Stop B210, Los Alamos, New Mexico 87545 Email: jimlouckLANL.GOV Communicated by the Managing Editors Received March 5, 1997
We use the principle of inclusion and exclusion to enumerate labeled cubic graphs, without resort to the superposition theory of Read. This work was motivated by the cubic array representation of cubic graphs in the studies of generating functions of 3n& j coefficients in angular momentum theory. 1999 Academic Press
1. INTRODUCTION The enumeration of labeled regular graphs without loops or multiple edges has been extensively studied in enumerative combinatorics since the first solution given by Read [11, 12]; see also [8] in terms of the theory of superposition. Although an expression has been found by Read for the number of regular graphs of higher degree, it is quite involved to derive explicit formulas. Relevant studies have been carried out in [1, 4, 6, 7, 10, 14, 16, 17]. For the case of cubic graphs, the following formula is given by Read [12]. Theorem 1. given by C 2n =
The number C 2n of labeled cubic graphs on 2n vertices is (&1) i (6j&2i)! 6 i (&1) k (2n)! . : n j : 6 i, j (3j&i)! (2j&i)! (n& j)! 48 k (i&2k)! k!
(1)
151 0097-316599 30.00 Copyright 1999 by Academic Press All rights of reproduction in any form reserved.
152
NOTE
From now on, all graphs are assumed to be simple (without loops and multiple edges), and to have labeled vertices as well as labeled edges. Specifically, a cubic graph on 2n vertices is meant to have vertices 1, 2, ..., 2n and edges 1, 2, ..., 3n. Given a graph G=(V, E) with V=[v 1 , v 2 , ..., v n ] and E=[e 1 , e 2 , ..., e m ], the incidence matrix of G is defined as an m_n (0, 1)-matrix A=(a ij ) given by a ij =
{
1, 0,
if e i is incident to v j , otherwise.
The incidence matrix A of a cubic graph on 2n vertices has the following properties: (1) A has 3n rows and 2n columns; (2) A has row sums two and column sums three; (3) A does not contain any 2_2 submatrix consisting of all ones. Moreover, one sees that (0, 1) matrix with properties (1)(3) defines a cubic graph with labeled vertices and edges. This formulation enables us to employ the principle of inclusion and exclusion to enumerate incidence matrices of cubic graphs. Our method also applies to the enumeration of r-regular graphs, for small r. Our approach was motivated by the cubic array representation of cubic graphs, an equivalent form to the incidence matrices, in the studies of generating functions of 3n& j coefficients in angular momentum theory [2, 3]. We need some background on symmetric functions. If we disregard the pattern restrictions for the moment, then all possibilities of placing three 1's in each column and distributed into m rows is expressed by the terms appearing in the following elementary symmetric function: e 3(z 1 , z 2 , ..., z m )=
:
z i1 z i2 z i3 .
1 i1
The power sum symmetric function is defined by p i (z 1 , z 2 , ..., z m )=z i1 +z i2 + } } } +z im , and e 3 is expressed in terms of the p i by the well-known relation [5, 9]: e 3 = 16 ( p 31 &3p 1 p 2 +2p 3 ). The following lemma is easy, which also follows from a more general theorem of Gessel [6]. We present a proof for completeness. For a monomial z k11 z k22 } } } z kmm , and a polynomial f (z 1 , z 2 , ..., z m ), we adopt the usual notation [z k11 z k22 } } } z kmm ] f (z 1 , z 2 , ..., z m ) for the coefficient of z k11 z k22 } } } z kmm in f (z 1 , z 2 , ..., z m ).
153
NOTE
Lemma 2. We have [z 21 z 22 } } } z 2m ] p 2(m&r) p r2 =m! (2m&2r&1)!!. 1 Proof.
p r2 in the following form: Write p 2(m&r) 1
(z 21 +z 22 + } } } +z 2m ) } } } (z 21 +z 22 + } } } +z 2m )(z 1 +z 2 + } } } +z m ) 2(m&r). To obtain the term z 21 z 22 } } } z 2m in the expansion of the above product, one picks up z 2i1 in the first factor, z 2i2 in the second factor,..., and z 2ir in the r-th factor, in which case z i1 , z i2 , ... have to be omitted in the last factor (z 1 +z 2 + } } } z m ) 2(m&r). It follows that the coefficient of z 21 z 22 } } } z 2m in the expansion equals (m) r
[2(m&r)]! , 2 m&r
where (m) r =m(m&1) } } } (m&r+1) is the lower factorial. The above number can be easily simplified to m! (2m&2r&1)!!, where for an odd positive integer k, we adopt the notation k!! for the double factorial defined as k!!=k } (k&2) } } } 3 } 1 with the convention (&1)!!=1. K We are now ready to use the inclusion-exclusion principle for the enumeration of incidence matrices of cubic graphs. Let M 2n be the set of (3n)_(2n) (0, 1)-matrices with row sums 2 and column sums 3. Let I=[i 1 , i 2 ] be a 2-subset of [3n] and J=[ j 1 , j 2 ] a 2-subset of [2n], where [k]=[1, 2, ..., k] for any positive integer k. Let P I, J be the set of matrices A in M 2n such that elements in the submatrix A[I, J] of A consisting of the rows i 1 and i 2 and columns j 1 and j 2 are all ones. The number of cubic graphs on 2n vertices is then given by
}
}
|M 2n | & . P I, J . I, J
Recall the following formula of inclusion-exclusion for a family of subsets Si of a set S [13, 15]: |S| & |S 1 _ S 2 _ } } } | = |S| &: |S i | i
+ : |S i & S j | & : i< j
We now make an easy observation.
i< j
|S i & S j & S k | & } } } .
(2)
154
NOTE
Lemma 3. Let I 1 , I 2 be 2-subsets of [3n] and J 1 , J 2 be 2-subsets of [2n] such that (I 1 , J 1 ){(I 2 , J 2 ). Suppose that P I1 , J1 and P I2 , J2 have a matrix in common, then we have either I 1 and I 2 are disjoint and J 1 and J 2 are disjoint, or J 1 =J 2 and I 1 and I 2 have exactly one element in common. Proof. Because of the row sum and column sum conditions, it is impossible that J 1 and J 2 can have exactly one element in common given that P I1 , J1 and P I2 , J2 have a matrix in common. Now assume that J 1 and J 2 are disjoint. From the row sum condition, it is clear that I 1 and I 2 are disjoint. When J 1 =J 2 , I 1 and I 2 must have exactly one element in common because of the column sum condition. K Note that when I 1 and I 2 have exactly one element in common, then we may write P I1 _ I2 , J =P I1 , J & P I2 , J . Let I$=[i 1 , i 2 , i 3 ]. Then there are three ways to write P I$, J as an intersection of P I1 , J & P I2 , J corresponding to the following possible choices for (I 1 , I 2 ): ([i 1 , i 2 ], [i 1 , i 3 ]),
([i 1 , i 2 ], [i 2 , i 3 ]),
([i 1 , i 3 ], [i 2 , i 3 ]).
Moreover, there is only one way to write P I$, J as an intersection of three P I, J 's for |I| =2 and I/I$. Proof of Theorem 1. By the principle of inclusion and exclusion, we need to compute the sum over the cardinalities of the unions of all possible k sets out of P I, J 's. By the above Lemma 3, we need to compute the number W k, t defined by W k, t =: |P I1 , J1 & } } } & P Ik , Jk & P R1 , S1 & } } } & P Rt , St |,
(3)
where I 1 , I 2 , ..., I k , R 1 , R 2 , ..., R t are disjoint subsets of [3n] such that |I 1 | = } } } = |I k | =2 and |R 1 | = } } } = |R t | =3, J 1 , ..., J k , S 1 , ..., S t are disjoint subsets of [2n]. By the principle of inclusion and exclusion, the number of cubic graphs is expressed in terms of the integers W k, t by : (&1) k 2 t W k, t . k, t
Given k and t as in (3), the number of ways to form the set [J 1 , J 2 , ..., J k ] equals 2n
\2k+ (2k&1)!!.
155
NOTE
After the set [J 1 , J 2 , ..., J k ] of indices are chosen, we may choose the set [I 1 , I 2 , ..., I k ] and then form the pairs [(I 1 , J 1 ), ..., (I k , J k )]. The number of ways to effect this equals 3n
\2k+ (2k&1)!! k!.
(4)
Given the pairs [(I 1 , J 1 ), ..., (I k , J k )], the number of ways to choose [S 1 , S 2 , ..., S t ] equals
\
2n&2k (2t&1)!!. 2t
+
Then the number of ways to choose R 1 , R 2 , ..., R t and to form the pairs [(R 1 , S 1 ), ..., (R t , S t )] equals
\
3n&2k (3t)! . 3t (3!) t
+
Deleting the rows corresponding to I 1 , I 2 , ..., I k and R 1 , R 2 , ..., R t , and the columns corresponding to S 1 , S 2 , ..., S t , we obtain matrices having 3n&2k&3t rows and 2n&2t columns with row sums 2, and 2k columns with column sums 1, and 2n&2k&2t columns with column sums 3. The number of such matrices equals the coefficient of z 21 z 22 } } } z 23n&2k&3t in 2n&2k&2t . Denote this number by F 2n, k, t . Let m=3n&2k&3t. Then p 2k 1 e3 we have F 2n, k, t =[z 21 } } } z 2m ](z 1 + } } } +z m ) 2k e 3(z 1 , ..., z m ) 2n&2k&2t =[z 21 } } } z 2m ] =[z 21 } } } z 2m ] _ =
\
2k 1 2n&2k&2t
p
6
1 6 2n&2k&2t
( p 31 &3p 1 p 2 +2p 3 ) 2n&2k&2t 2n&2k&2t
:
(&1) i
i=0
2n&2k&2t 2n&2k&2t&i 2n&2t+2i 2n&2k&2t&i 3 p1 p2 i
+
(3n&2k&3t)! 2n&2k&2t : (&1) i 3 2n&2k&2t&i 6 2n&2k&2t i=0 _
\
2n&2k&2t (2n&2t+2i&1)!!. i
+
156
NOTE
Observe that the term p 3 in e 3 cannot contribute to F 2n, k, t and can be deleted from the expansion. Using the formula (2) of inclusion and exclusion and synthesizing the number of ways to form the indices in (3), we obtain the following formula for the number of cubic graphs with 2n labeled vertices and 3n labeled edges for n 2: (3n)! (2n)! (&1) k 2 t 2n&2k&2t (&1) i (2n&2t+2i&1)!! : . : 4n 3 t k! t! 3 i i! (2n&2k&2t&i)! i=0 k+t n
(5)
Removing the factor (3n)! in (5), we obtain the number of cubic graphs with 2n labeled vertices and unlabeled edges. Replacing i by 2n&2k&2t&i in the second summation in (5), it can be rewritten as 2n&2k&2t
: i=0
(&1) i (6n&4k&6t&2i&1)!! . 3 2n&2k&2t&i i! (2n&2k&2t&i)!
Replacing i by i&2k, and n&t by j, then (5) becomes (1) by dropping the factor (3n)!. K
ACKNOWLEDGMENTS This work was done under the auspices of the US Department of Energy, the National Science Foundation and MCSEC of China. We thank Q. H. Hou and Yi Wang for valuable comments. We also thank the referee for valuable suggestions.
REFERENCES 1. E. A. Bender and E. R. Canfield, The asymptotic number of labelled graphs with given degree sequence, J. Combin. Theory Ser. A 24 (1978), 296307. 2. L. C. Biedenharn, W. Y. C. Chen, M. A. Lohe, and J. D. Louck, The role of 3n& j coefficients in SU(3), in ``Symmetry and Structural Properties of Condensed Matter'' (T. Lulek, W. Florek, and S. Walcerz, Eds.), pp. 150182, Proc. Third SSCPM, World Scientific, Singapore, 1995. 3. L. C. Biedenharn and J. D. Louck, ``Angular Momentum Theory in Quantum Physics,'' Encyclopedia of Mathematics and Its Applications (G.-C. Rota, Ed.), Vol. 8, Cambridge Univ. Press, Cambridge, UK, 1981. 4. B. Bollobas, A probabilistic proof of an asymptotic formula for the number of labelled regular graphs, Europ. J. Combin. 1 (1980), 311316. 5. F. Bonetti, G.-C. Rota, D. Senato, and A. M. Venezia, On the foundations of combinatorial theory, X, A categorical setting for symmetric functions, Stud. Appl. Math. 86 (1992), 129. 6. I. M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), 257285.
NOTE
157
7. I. P. Goulden and D. M. Jackson, Labelled graphs with small vertex degrees and P-recursiveness, SIAM J. Algebra. Discr. Methods 7 (1986), 6066. 8. F. Harary and E. M. Palmer, ``Graphical Enumeration,'' Academic Press, New York, 1973. 9. I. G. Macdonald, ``Symmetric Functions and Hall Polynomials,'' Oxford Univ. Press, Oxford, 1979. 10. B. D. McKay, E. M. Palmer, R. C. Read, and R. W. Robinson, The asymptotic number of claw-free cubic graphs, preprint, 1997. 11. R. C. Read, The enumeration of locally restricted graphs, I, J. London Math. Soc. 34 (1959), 417436. 12. R. C. Read, The enumeration of locally restricted graphs, II, J. London Math. Soc. 35 (1960), 344351. 13. G.-C. Rota, On the foundations of combinatorial theory, I, Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie 2 (1964), 340368. 14. R. P. Stanley, Differentiably finite power series, Europ. J. Combin. 1 (1980), 175188. 15. R. P. Stanley, ``Enumerative Combinatorics,'' Vol. I, Wadsworth 6 BrooksCole, Monterey, CA, 1986. 16. N. C. Wormald, Enumeration of labelled graphs, II, Cubic graphs with a given connectivity, J. London Math. Soc. 20 (1979), 17. 17. D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990), 321368.
NOTE Enumeration of Cubic Graphs by InclusionExclusion William Y. C. Chen Los Alamos National Laboratory, T-7, Mail Stop B284, Los Alamos, New Mexico 87545, and Center for Combinatorics, Nankai University, Tianjin 300071, People's Republic of China E-mail: chent7.lanl.gov, chenpublic.tpt.tj.cn
and James D. Louck Los Alamos National Laboratory, T-7, Mail Stop B210, Los Alamos, New Mexico 87545 Email: jimlouckLANL.GOV Communicated by the Managing Editors Received March 5, 1997
We use the principle of inclusion and exclusion to enumerate labeled cubic graphs, without resort to the superposition theory of Read. This work was motivated by the cubic array representation of cubic graphs in the studies of generating functions of 3n& j coefficients in angular momentum theory. 1999 Academic Press
1. INTRODUCTION The enumeration of labeled regular graphs without loops or multiple edges has been extensively studied in enumerative combinatorics since the first solution given by Read [11, 12]; see also [8] in terms of the theory of superposition. Although an expression has been found by Read for the number of regular graphs of higher degree, it is quite involved to derive explicit formulas. Relevant studies have been carried out in [1, 4, 6, 7, 10, 14, 16, 17]. For the case of cubic graphs, the following formula is given by Read [12]. Theorem 1. given by C 2n =
The number C 2n of labeled cubic graphs on 2n vertices is (&1) i (6j&2i)! 6 i (&1) k (2n)! . : n j : 6 i, j (3j&i)! (2j&i)! (n& j)! 48 k (i&2k)! k!
(1)
151 0097-316599 30.00 Copyright 1999 by Academic Press All rights of reproduction in any form reserved.
152
NOTE
From now on, all graphs are assumed to be simple (without loops and multiple edges), and to have labeled vertices as well as labeled edges. Specifically, a cubic graph on 2n vertices is meant to have vertices 1, 2, ..., 2n and edges 1, 2, ..., 3n. Given a graph G=(V, E) with V=[v 1 , v 2 , ..., v n ] and E=[e 1 , e 2 , ..., e m ], the incidence matrix of G is defined as an m_n (0, 1)-matrix A=(a ij ) given by a ij =
{
1, 0,
if e i is incident to v j , otherwise.
The incidence matrix A of a cubic graph on 2n vertices has the following properties: (1) A has 3n rows and 2n columns; (2) A has row sums two and column sums three; (3) A does not contain any 2_2 submatrix consisting of all ones. Moreover, one sees that (0, 1) matrix with properties (1)(3) defines a cubic graph with labeled vertices and edges. This formulation enables us to employ the principle of inclusion and exclusion to enumerate incidence matrices of cubic graphs. Our method also applies to the enumeration of r-regular graphs, for small r. Our approach was motivated by the cubic array representation of cubic graphs, an equivalent form to the incidence matrices, in the studies of generating functions of 3n& j coefficients in angular momentum theory [2, 3]. We need some background on symmetric functions. If we disregard the pattern restrictions for the moment, then all possibilities of placing three 1's in each column and distributed into m rows is expressed by the terms appearing in the following elementary symmetric function: e 3(z 1 , z 2 , ..., z m )=
:
z i1 z i2 z i3 .
1 i1
The power sum symmetric function is defined by p i (z 1 , z 2 , ..., z m )=z i1 +z i2 + } } } +z im , and e 3 is expressed in terms of the p i by the well-known relation [5, 9]: e 3 = 16 ( p 31 &3p 1 p 2 +2p 3 ). The following lemma is easy, which also follows from a more general theorem of Gessel [6]. We present a proof for completeness. For a monomial z k11 z k22 } } } z kmm , and a polynomial f (z 1 , z 2 , ..., z m ), we adopt the usual notation [z k11 z k22 } } } z kmm ] f (z 1 , z 2 , ..., z m ) for the coefficient of z k11 z k22 } } } z kmm in f (z 1 , z 2 , ..., z m ).
153
NOTE
Lemma 2. We have [z 21 z 22 } } } z 2m ] p 2(m&r) p r2 =m! (2m&2r&1)!!. 1 Proof.
p r2 in the following form: Write p 2(m&r) 1
(z 21 +z 22 + } } } +z 2m ) } } } (z 21 +z 22 + } } } +z 2m )(z 1 +z 2 + } } } +z m ) 2(m&r). To obtain the term z 21 z 22 } } } z 2m in the expansion of the above product, one picks up z 2i1 in the first factor, z 2i2 in the second factor,..., and z 2ir in the r-th factor, in which case z i1 , z i2 , ... have to be omitted in the last factor (z 1 +z 2 + } } } z m ) 2(m&r). It follows that the coefficient of z 21 z 22 } } } z 2m in the expansion equals (m) r
[2(m&r)]! , 2 m&r
where (m) r =m(m&1) } } } (m&r+1) is the lower factorial. The above number can be easily simplified to m! (2m&2r&1)!!, where for an odd positive integer k, we adopt the notation k!! for the double factorial defined as k!!=k } (k&2) } } } 3 } 1 with the convention (&1)!!=1. K We are now ready to use the inclusion-exclusion principle for the enumeration of incidence matrices of cubic graphs. Let M 2n be the set of (3n)_(2n) (0, 1)-matrices with row sums 2 and column sums 3. Let I=[i 1 , i 2 ] be a 2-subset of [3n] and J=[ j 1 , j 2 ] a 2-subset of [2n], where [k]=[1, 2, ..., k] for any positive integer k. Let P I, J be the set of matrices A in M 2n such that elements in the submatrix A[I, J] of A consisting of the rows i 1 and i 2 and columns j 1 and j 2 are all ones. The number of cubic graphs on 2n vertices is then given by
}
}
|M 2n | & . P I, J . I, J
Recall the following formula of inclusion-exclusion for a family of subsets Si of a set S [13, 15]: |S| & |S 1 _ S 2 _ } } } | = |S| &: |S i | i
+ : |S i & S j | & : i< j
We now make an easy observation.
i< j
|S i & S j & S k | & } } } .
(2)
154
NOTE
Lemma 3. Let I 1 , I 2 be 2-subsets of [3n] and J 1 , J 2 be 2-subsets of [2n] such that (I 1 , J 1 ){(I 2 , J 2 ). Suppose that P I1 , J1 and P I2 , J2 have a matrix in common, then we have either I 1 and I 2 are disjoint and J 1 and J 2 are disjoint, or J 1 =J 2 and I 1 and I 2 have exactly one element in common. Proof. Because of the row sum and column sum conditions, it is impossible that J 1 and J 2 can have exactly one element in common given that P I1 , J1 and P I2 , J2 have a matrix in common. Now assume that J 1 and J 2 are disjoint. From the row sum condition, it is clear that I 1 and I 2 are disjoint. When J 1 =J 2 , I 1 and I 2 must have exactly one element in common because of the column sum condition. K Note that when I 1 and I 2 have exactly one element in common, then we may write P I1 _ I2 , J =P I1 , J & P I2 , J . Let I$=[i 1 , i 2 , i 3 ]. Then there are three ways to write P I$, J as an intersection of P I1 , J & P I2 , J corresponding to the following possible choices for (I 1 , I 2 ): ([i 1 , i 2 ], [i 1 , i 3 ]),
([i 1 , i 2 ], [i 2 , i 3 ]),
([i 1 , i 3 ], [i 2 , i 3 ]).
Moreover, there is only one way to write P I$, J as an intersection of three P I, J 's for |I| =2 and I/I$. Proof of Theorem 1. By the principle of inclusion and exclusion, we need to compute the sum over the cardinalities of the unions of all possible k sets out of P I, J 's. By the above Lemma 3, we need to compute the number W k, t defined by W k, t =: |P I1 , J1 & } } } & P Ik , Jk & P R1 , S1 & } } } & P Rt , St |,
(3)
where I 1 , I 2 , ..., I k , R 1 , R 2 , ..., R t are disjoint subsets of [3n] such that |I 1 | = } } } = |I k | =2 and |R 1 | = } } } = |R t | =3, J 1 , ..., J k , S 1 , ..., S t are disjoint subsets of [2n]. By the principle of inclusion and exclusion, the number of cubic graphs is expressed in terms of the integers W k, t by : (&1) k 2 t W k, t . k, t
Given k and t as in (3), the number of ways to form the set [J 1 , J 2 , ..., J k ] equals 2n
\2k+ (2k&1)!!.
155
NOTE
After the set [J 1 , J 2 , ..., J k ] of indices are chosen, we may choose the set [I 1 , I 2 , ..., I k ] and then form the pairs [(I 1 , J 1 ), ..., (I k , J k )]. The number of ways to effect this equals 3n
\2k+ (2k&1)!! k!.
(4)
Given the pairs [(I 1 , J 1 ), ..., (I k , J k )], the number of ways to choose [S 1 , S 2 , ..., S t ] equals
\
2n&2k (2t&1)!!. 2t
+
Then the number of ways to choose R 1 , R 2 , ..., R t and to form the pairs [(R 1 , S 1 ), ..., (R t , S t )] equals
\
3n&2k (3t)! . 3t (3!) t
+
Deleting the rows corresponding to I 1 , I 2 , ..., I k and R 1 , R 2 , ..., R t , and the columns corresponding to S 1 , S 2 , ..., S t , we obtain matrices having 3n&2k&3t rows and 2n&2t columns with row sums 2, and 2k columns with column sums 1, and 2n&2k&2t columns with column sums 3. The number of such matrices equals the coefficient of z 21 z 22 } } } z 23n&2k&3t in 2n&2k&2t . Denote this number by F 2n, k, t . Let m=3n&2k&3t. Then p 2k 1 e3 we have F 2n, k, t =[z 21 } } } z 2m ](z 1 + } } } +z m ) 2k e 3(z 1 , ..., z m ) 2n&2k&2t =[z 21 } } } z 2m ] =[z 21 } } } z 2m ] _ =
\
2k 1 2n&2k&2t
p
6
1 6 2n&2k&2t
( p 31 &3p 1 p 2 +2p 3 ) 2n&2k&2t 2n&2k&2t
:
(&1) i
i=0
2n&2k&2t 2n&2k&2t&i 2n&2t+2i 2n&2k&2t&i 3 p1 p2 i
+
(3n&2k&3t)! 2n&2k&2t : (&1) i 3 2n&2k&2t&i 6 2n&2k&2t i=0 _
\
2n&2k&2t (2n&2t+2i&1)!!. i
+
156
NOTE
Observe that the term p 3 in e 3 cannot contribute to F 2n, k, t and can be deleted from the expansion. Using the formula (2) of inclusion and exclusion and synthesizing the number of ways to form the indices in (3), we obtain the following formula for the number of cubic graphs with 2n labeled vertices and 3n labeled edges for n 2: (3n)! (2n)! (&1) k 2 t 2n&2k&2t (&1) i (2n&2t+2i&1)!! : . : 4n 3 t k! t! 3 i i! (2n&2k&2t&i)! i=0 k+t n
(5)
Removing the factor (3n)! in (5), we obtain the number of cubic graphs with 2n labeled vertices and unlabeled edges. Replacing i by 2n&2k&2t&i in the second summation in (5), it can be rewritten as 2n&2k&2t
: i=0
(&1) i (6n&4k&6t&2i&1)!! . 3 2n&2k&2t&i i! (2n&2k&2t&i)!
Replacing i by i&2k, and n&t by j, then (5) becomes (1) by dropping the factor (3n)!. K
ACKNOWLEDGMENTS This work was done under the auspices of the US Department of Energy, the National Science Foundation and MCSEC of China. We thank Q. H. Hou and Yi Wang for valuable comments. We also thank the referee for valuable suggestions.
REFERENCES 1. E. A. Bender and E. R. Canfield, The asymptotic number of labelled graphs with given degree sequence, J. Combin. Theory Ser. A 24 (1978), 296307. 2. L. C. Biedenharn, W. Y. C. Chen, M. A. Lohe, and J. D. Louck, The role of 3n& j coefficients in SU(3), in ``Symmetry and Structural Properties of Condensed Matter'' (T. Lulek, W. Florek, and S. Walcerz, Eds.), pp. 150182, Proc. Third SSCPM, World Scientific, Singapore, 1995. 3. L. C. Biedenharn and J. D. Louck, ``Angular Momentum Theory in Quantum Physics,'' Encyclopedia of Mathematics and Its Applications (G.-C. Rota, Ed.), Vol. 8, Cambridge Univ. Press, Cambridge, UK, 1981. 4. B. Bollobas, A probabilistic proof of an asymptotic formula for the number of labelled regular graphs, Europ. J. Combin. 1 (1980), 311316. 5. F. Bonetti, G.-C. Rota, D. Senato, and A. M. Venezia, On the foundations of combinatorial theory, X, A categorical setting for symmetric functions, Stud. Appl. Math. 86 (1992), 129. 6. I. M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), 257285.
NOTE
157
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