**6. Enumerating regular graphs**

As we mentioned in Section 6, some enumerating problems can be solved easily using generating tools for obtaining a closed formula. Some other problems are more hard to deal with for obtaining a closed mathematical expression, but we can resort in such a case to the asymptotic approximation of the coefficients of the power series [22–25]. It was also mentioned in Section 6 that there are some graph enumerating problems where the nodes are labeled, and in such a case the use of the *exponential generating functions* is well adapted for these kinds of problems. The other case of graph enumerating problems is when we are dealing with graphs whose nodes do not have an assigned label. Then, we can resort in such case to Pólya's enumerating method [7, 13], and the best choice is to use *ordinary generating functions*. It should also be mentioned that the edges of the graphs to be enumerated can be directed or undirected.

One of the seminal articles of enumerating graphs is [26], where a fundamental theorem was proven in the theory of random graphs on *n* unlabeled nodes and with a given number *q* of edges.

In [26], the authors obtained a necessary and sufficient condition for relating asymptotically the number of unlabeled graphs with *n* nodes and *q* edges with the number of labeled graphs with *n* nodes and *q* edges. Let *Tnq* be the number of different graphs with *n* nodes and *q* edges, *Fnq* the corresponding to number of

$$\text{Ilabel graphs, } N = \frac{n(n-1)}{2} \text{ the possible edges, and } F\_{nq} = \binom{N}{q} = \frac{N!}{q!(N-q)!}. \text{ The result}$$

obtained in [26] can be stated as the following theorem:

Theorem 1.7 The necessary and sufficient conditions that

$$T\_{nq} \sim \frac{F\_{nq}}{n!} \tag{41}$$

each row, **r** a vector over ½ �¼ *d* f g 0*;* 1*;* …*d* , and *G M*ð Þ *;r; t* the number of *n* � *n*

*G M*ð Þ� *;* **<sup>r</sup>***; <sup>t</sup> T f* ð Þ *; <sup>δ</sup> <sup>e</sup><sup>ε</sup>a*�*<sup>b</sup>*

*, b* <sup>¼</sup> <sup>P</sup>

0

BB@

and *T f* ð Þ *; δ* being the number of involutions on [1, *f*] such that no element in some

Three years later, the article [28] appeared giving a different approach of [27] allowing for obtaining a more general asymptotic formula without reference to an exact formula. The asymptotic result obtained by Bela Bollobás in [28] for enumerating labeled regular graphs is proven by a probabilistic method. This result can be stated as follows. Let Δ and *n* be natural numbers such that Δ*n* ¼ 2*m* is even and

graph *G*. Then, as *n* ! ∞, the number of labeled Δ-regular graphs on *n* vertices is

�*λ*�*λ*<sup>2</sup> ð Þ 2*m* !

constant but also for Δ growing slowly as *n* ! ∞ and summarized this in the

The authors of [27] affirm that the asymptotic Formula (44) holds not only for Δ

�*λ*�*λ*<sup>2</sup> ð Þ 2*m <sup>m</sup>* 2*<sup>m</sup>* Q*<sup>n</sup>*

In the next year, the author on [27] extended this result to the case of unlabeled graphs in [29]. The result of Theorem 1.9 extended for the case of unlabeled graphs

*<sup>m</sup>*!2*m*ð Þ <sup>Δ</sup>! *<sup>m</sup>*, then

*<sup>n</sup>*¼<sup>1</sup> <sup>M</sup>ð Þ� *<sup>n</sup>; <sup>z</sup>* ½ � <sup>0</sup>*; <sup>d</sup> <sup>n</sup>* ð Þ as *<sup>f</sup>* ! <sup>∞</sup>, where

*mij*¼0*,i* <*j*

2, where *n* is the number of vertices and *m* is the number of edges of the

*rirj* <sup>þ</sup> <sup>P</sup> *i*

<sup>Q</sup> *ri*! *:* (43)

*ri* 2 � �

1

CCA*, <sup>δ</sup>* <sup>¼</sup> <sup>P</sup>

*<sup>i</sup>*¼<sup>1</sup> *di* <sup>¼</sup> <sup>2</sup>*<sup>m</sup>* even.

<sup>1</sup> satisfies

*<sup>i</sup>*¼<sup>1</sup> *di*! � � *,* (45)

*mij*¼<sup>0</sup>*ri*

*f*

*<sup>m</sup>*!2*<sup>m</sup>*ð Þ <sup>Δ</sup>! *<sup>m</sup> ,* (44)

<sup>2</sup> � 1 and *m* ≥ maxf g *ε*Δ*n;*ð Þ 1 þ *ε n* for some *ε*>0. Then, the

symmetric matrices *gij* � � over ½�¼*<sup>t</sup>* f g <sup>0</sup>*;* <sup>1</sup>*;* …*<sup>t</sup>* such that

i. *gij* ¼ 0 if *mij* ¼ 0.

*Combinatorial Enumeration of Graphs DOI: http://dx.doi.org/10.5772/intechopen.88805*

Theorem 1.8 For given *d, t* and *z*,

Uniformly, for ð Þ *<sup>M</sup>;* **<sup>r</sup>** ∈ ∪ <sup>∞</sup>

2

þ

1

CCA

specified set of size *δ* is fixed.

*ri, ε* ¼ �1 if *t* ¼ 1 and þ 1 if *t*>1, for

*ri* 2 � �

1

CCA

*e*

Theorem 1.9 Let *d*<sup>1</sup> ≥ *d*<sup>2</sup> ≥ …*dn* be natural numbers with P*<sup>n</sup>*

number *<sup>L</sup>*ð Þ **<sup>d</sup>** of labeled graphs with degree sequence **<sup>d</sup>** <sup>¼</sup> ð Þ *di <sup>n</sup>*

*L*ð Þ� **d** *e*

*di* 2 � �*:*

can be summarized in the following theorem. Theorem 1.10 If <sup>Δ</sup> <sup>≥</sup>3 and *<sup>L</sup>*<sup>Δ</sup> <sup>¼</sup> *<sup>e</sup>*�*λ*�*λ*<sup>2</sup> ð Þ <sup>2</sup>*<sup>m</sup>* !

*f*

P *mij*

0

BB@

ii. P *j gij* ¼ *ri*.

*<sup>f</sup>* <sup>¼</sup> <sup>P</sup> *i*

> P *i ri* 2 � �

0

BB@

<sup>Δ</sup> <sup>≤</sup>ð Þ 2 log *<sup>n</sup>* <sup>1</sup>

asymptotic to

where *<sup>λ</sup>* <sup>¼</sup> ð Þ <sup>Δ</sup>�<sup>1</sup>

following theorem.

where *<sup>λ</sup>* <sup>¼</sup> <sup>1</sup>

**237**

2*m* P*<sup>n</sup> i*¼1

Suppose <sup>Δ</sup> <sup>¼</sup> *<sup>d</sup>*<sup>1</sup> <sup>≤</sup>ð Þ 2 log *<sup>n</sup>* <sup>1</sup>

<sup>2</sup> .

*f*

*a* ¼

as *n* ! ∞ is that

$$
\min(q, N - q)/n - (\log n/2) \to \infty.\tag{42}
$$

The formal result expressed in Theorem 1.7 for unlabeled graphs is a starting point on the enumeration of regular graphs because it allows for enumerating those unlabeled graphs that have some number of edges. In fact, the author of [26] proved that if a graph with ∣*E*∣ ¼ *E n*ð Þ edges, where *n* is the number of vertices or order of such a graph, has no isolated vertices or two vertices of degree *n* � 1, then the number of unlabeled graphs of order *n* and number of edges ∣*E*∣ divided by the number unlabeled graphs is asymptotic to *n*!.

Another interesting article on asymptotic enumeration of labeled graphs having a given degree sequence was [27]. The authors of [27] obtained their asymptotic result for *n* � *n* symmetric matrices subject to the following constraints:

i. Each row sum is specified and bounded.


The authors of [27] mentioned that their results can be interpreted in terms of incidence matrices for labeled graphs. The results of [27] can be stated as follows. Let Mð Þ *n; z* be the set of all *n* � *n* symmetric 0ð Þ *;* 1 matrices with at most *z* zeroes in each row, **r** a vector over ½ �¼ *d* f g 0*;* 1*;* …*d* , and *G M*ð Þ *;r; t* the number of *n* � *n* symmetric matrices *gij* � � over ½�¼*<sup>t</sup>* f g <sup>0</sup>*;* <sup>1</sup>*;* …*<sup>t</sup>* such that

i.g.  $\mathbf{j}\_{i\dot{j}} = \mathbf{0}$  if  $m\_{i\dot{j}} = \mathbf{0}$ .

ili.  $\sum\_{j} \mathbf{g}\_{i\dot{j}} = r\_{i}$ .

**6. Enumerating regular graphs**

*Probability, Combinatorics and Control*

can be directed or undirected.

a given number *q* of edges.

labeled graphs, *<sup>N</sup>* <sup>¼</sup> *n n*ð Þ �<sup>1</sup>

as *n* ! ∞ is that

As we mentioned in Section 6, some enumerating problems can be solved easily using generating tools for obtaining a closed formula. Some other problems are more hard to deal with for obtaining a closed mathematical expression, but we can resort in such a case to the asymptotic approximation of the coefficients of the power series [22–25]. It was also mentioned in Section 6 that there are some graph enumerating problems where the nodes are labeled, and in such a case the use of the *exponential generating functions* is well adapted for these kinds of problems. The other case of graph enumerating problems is when we are dealing with graphs whose nodes do not have an assigned label. Then, we can resort in such case to Pólya's enumerating method [7, 13], and the best choice is to use *ordinary generating functions*. It should also be mentioned that the edges of the graphs to be enumerated

One of the seminal articles of enumerating graphs is [26], where a fundamental theorem was proven in the theory of random graphs on *n* unlabeled nodes and with

In [26], the authors obtained a necessary and sufficient condition for relating asymptotically the number of unlabeled graphs with *n* nodes and *q* edges with the number of labeled graphs with *n* nodes and *q* edges. Let *Tnq* be the number of different graphs with *n* nodes and *q* edges, *Fnq* the corresponding to number of

<sup>2</sup> the possible edges, and *Fnq* <sup>¼</sup> *<sup>N</sup>*

*Tnq* � *Fnq*

The formal result expressed in Theorem 1.7 for unlabeled graphs is a starting point on the enumeration of regular graphs because it allows for enumerating those unlabeled graphs that have some number of edges. In fact, the author of [26] proved that if a graph with ∣*E*∣ ¼ *E n*ð Þ edges, where *n* is the number of vertices or order of such a graph, has no isolated vertices or two vertices of degree *n* � 1, then the number of unlabeled graphs of order *n* and number of edges ∣*E*∣ divided by the

Another interesting article on asymptotic enumeration of labeled graphs having a given degree sequence was [27]. The authors of [27] obtained their asymptotic

The authors of [27] mentioned that their results can be interpreted in terms of incidence matrices for labeled graphs. The results of [27] can be stated as follows. Let Mð Þ *n; z* be the set of all *n* � *n* symmetric 0ð Þ *;* 1 matrices with at most *z* zeroes in

result for *n* � *n* symmetric matrices subject to the following constraints:

obtained in [26] can be stated as the following theorem:

number unlabeled graphs is asymptotic to *n*!.

i. Each row sum is specified and bounded.

iii. A specified *sparse* set of entries must be zero.

ii. The entries are bounded.

**236**

Theorem 1.7 The necessary and sufficient conditions that

*q* 

minð Þ *q; N* � *q =n* � ð Þ! log *n=*2 ∞*:* (42)

*<sup>n</sup>*! (41)

<sup>¼</sup> *<sup>N</sup>*! *q*!ð Þ *N*�*q* !

. The result

Theorem 1.8 For given *d, t* and *z*,

$$G(\mathbf{M}, \mathbf{r}, t) \sim \frac{T(f, \delta) \varepsilon^{\alpha - b}}{\prod r\_i!}. \tag{43}$$

Uniformly, for ð Þ *<sup>M</sup>;* **<sup>r</sup>** ∈ ∪ <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> <sup>M</sup>ð Þ� *<sup>n</sup>; <sup>z</sup>* ½ � <sup>0</sup>*; <sup>d</sup> <sup>n</sup>* ð Þ as *<sup>f</sup>* ! <sup>∞</sup>, where *<sup>f</sup>* <sup>¼</sup> <sup>P</sup> *i ri, ε* ¼ �1 if *t* ¼ 1 and þ 1 if *t*>1, for

$$a = \left(\frac{\sum\_{i} \binom{r\_i}{2}}{f}\right)^2 + \left(\frac{\sum\_{n\_{\vec{i}}} \binom{r\_i}{2}}{f}\right), b = \left(\sum\_{m\_{\vec{i}}=0, i$$

and *T f* ð Þ *; δ* being the number of involutions on [1, *f*] such that no element in some specified set of size *δ* is fixed.

Three years later, the article [28] appeared giving a different approach of [27] allowing for obtaining a more general asymptotic formula without reference to an exact formula. The asymptotic result obtained by Bela Bollobás in [28] for enumerating labeled regular graphs is proven by a probabilistic method. This result can be stated as follows. Let Δ and *n* be natural numbers such that Δ*n* ¼ 2*m* is even and <sup>Δ</sup> <sup>≤</sup>ð Þ 2 log *<sup>n</sup>* <sup>1</sup> 2, where *n* is the number of vertices and *m* is the number of edges of the graph *G*. Then, as *n* ! ∞, the number of labeled Δ-regular graphs on *n* vertices is asymptotic to

$$e^{-\lambda - \lambda^2} \frac{(2m)!}{m! 2^m (\Delta!)^m},\tag{44}$$

where *<sup>λ</sup>* <sup>¼</sup> ð Þ <sup>Δ</sup>�<sup>1</sup> <sup>2</sup> .

The authors of [27] affirm that the asymptotic Formula (44) holds not only for Δ constant but also for Δ growing slowly as *n* ! ∞ and summarized this in the following theorem.

Theorem 1.9 Let *d*<sup>1</sup> ≥ *d*<sup>2</sup> ≥ …*dn* be natural numbers with P*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup> *di* <sup>¼</sup> <sup>2</sup>*<sup>m</sup>* even. Suppose <sup>Δ</sup> <sup>¼</sup> *<sup>d</sup>*<sup>1</sup> <sup>≤</sup>ð Þ 2 log *<sup>n</sup>* <sup>1</sup> <sup>2</sup> � 1 and *m* ≥ maxf g *ε*Δ*n;*ð Þ 1 þ *ε n* for some *ε*>0. Then, the number *<sup>L</sup>*ð Þ **<sup>d</sup>** of labeled graphs with degree sequence **<sup>d</sup>** <sup>¼</sup> ð Þ *di <sup>n</sup>* <sup>1</sup> satisfies

$$L(\mathbf{d}) \sim \mathcal{e}^{-\lambda - \lambda^2} \frac{(2m)\_m}{\left\{ 2^m \prod\_{i=1}^n d\_i! \right\}},\tag{45}$$

where *<sup>λ</sup>* <sup>¼</sup> <sup>1</sup> 2*m* P*<sup>n</sup> i*¼1 *di* 2 � �*:*

In the next year, the author on [27] extended this result to the case of unlabeled graphs in [29]. The result of Theorem 1.9 extended for the case of unlabeled graphs can be summarized in the following theorem.

Theorem 1.10 If <sup>Δ</sup> <sup>≥</sup>3 and *<sup>L</sup>*<sup>Δ</sup> <sup>¼</sup> *<sup>e</sup>*�*λ*�*λ*<sup>2</sup> ð Þ <sup>2</sup>*<sup>m</sup>* ! *<sup>m</sup>*!2*m*ð Þ <sup>Δ</sup>! *<sup>m</sup>*, then

$$U\_{\Delta} \sim \frac{L\_{\Delta}}{n!} \sim e^{-\frac{(\Delta^2 - 1)}{4}} \frac{(2m)!}{2^m m!} \frac{(\Delta!)^{ - n}}{n!},\tag{46}$$

<sup>¼</sup> lim*n*!<sup>∞</sup>

ð Þ *<sup>c</sup>*1! *<sup>n</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffi

*e* � *c*2 1 �1 4 ffiffiffiffiffiffi 2*πc*<sup>1</sup> <sup>p</sup> *<sup>c</sup>*<sup>1</sup> ð Þ*<sup>e</sup> <sup>c</sup>*<sup>1</sup> ð Þ*<sup>n</sup>*

applying the approximation Stirling formula *<sup>n</sup>*! � ffiffiffiffiffiffiffiffi

<sup>¼</sup> lim*n*!<sup>∞</sup>

<sup>¼</sup> lim*<sup>n</sup>*!<sup>∞</sup>

applying the approximation Stirling formula

<sup>¼</sup> lim*<sup>n</sup>*!<sup>∞</sup>

then

*Combinatorial Enumeration of Graphs DOI: http://dx.doi.org/10.5772/intechopen.88805*

simplifying

and

we get

simplifying

**239**

*e* � *c*2 1 �1 4 ð Þ *<sup>c</sup>*1! *<sup>n</sup>*

> 2 *n n*ð Þ �1 2

2*πc*<sup>1</sup> p *c*<sup>1</sup>

> 2 *n n*ð Þ �1 2

1

ð Þ *<sup>c</sup>*<sup>1</sup> *<sup>c</sup>*1*<sup>n</sup> e c*2 1 þ4*c*1*n*þ1 4

> 2 *n n*ð Þ �1 2

<sup>2</sup>*π*ð Þ <sup>2</sup>*c*2*<sup>n</sup>* <sup>p</sup> ð Þ <sup>2</sup>*c*2*<sup>n</sup>*

<sup>2</sup>*π*ð Þ *<sup>c</sup>*2*<sup>n</sup>* <sup>p</sup> ð Þ *<sup>c</sup>*2*<sup>n</sup>*

2 *n n*ð Þ �1 2

ffiffi 2 p

> 2 *n n*ð Þ �1 2

given that *r*≥3, for which we assumed that *r* is a constant *c*1, and that *nr* ¼ 2*m*, then we have that *c*<sup>1</sup> ¼ 2*c*2, and replacing that, in (58), we can express it in terms of

*e* � �ð Þ <sup>2</sup>*c*2*<sup>n</sup>*

*e* � �ð Þ *<sup>c</sup>*2*<sup>n</sup>*

ffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>*π*ð Þ <sup>2</sup>*c*2*<sup>n</sup>* <sup>p</sup> <sup>2</sup>*c*<sup>2</sup> ð Þ*<sup>n</sup> e* � � <sup>2</sup>*c*<sup>2</sup> ð Þ*<sup>n</sup>*

<sup>2</sup>*c*2*<sup>n</sup>* ffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>*π*ð Þ *<sup>c</sup>*2*<sup>n</sup>* <sup>p</sup> *<sup>c</sup>*<sup>2</sup> ð Þ*<sup>n</sup> e* � � *<sup>c</sup>*<sup>2</sup> ð Þ*<sup>n</sup>*

> *c*2*n e* � �*<sup>c</sup>*2*<sup>n</sup>*

*c*1*n=*2 *e* � �*<sup>c</sup>*1*n=*<sup>2</sup>

ffiffiffiffiffiffi 2*πc*<sup>1</sup> p ð Þ*<sup>n</sup>*

ð Þ <sup>2</sup>*c*2*<sup>n</sup>* ! <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð Þ *<sup>c</sup>*2*<sup>n</sup>* ! <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

ð Þ *<sup>c</sup>*<sup>1</sup> *<sup>c</sup>*1*<sup>n</sup> e c*2 1 þ4*c*1*n*þ1 4

ffiffiffiffiffiffi 2*πc*<sup>1</sup> p ð Þ*<sup>n</sup> c c*1*n* <sup>1</sup> *<sup>e</sup> <sup>c</sup>*<sup>2</sup> 1 <sup>þ</sup>4*c*<sup>1</sup> ð Þ *<sup>n</sup>*þ<sup>1</sup> *<sup>=</sup>*<sup>4</sup>

> ffiffi 2 p

> > 2 *n n*ð Þ �1 2

ffiffiffiffiffiffi 2*πc*<sup>1</sup> p ð Þ*<sup>n</sup>*

<sup>¼</sup> lim*<sup>n</sup>*!<sup>∞</sup>

lim*n*!∞

*c*1, which is the regular degree *r* assumed as fixed; then, we get

ffiffiffiffiffiffi 2*πc*<sup>1</sup> p ð Þ*<sup>n</sup> c c*1*n* <sup>1</sup> *<sup>e</sup> <sup>c</sup>*<sup>2</sup> 1 <sup>þ</sup>4*c*<sup>1</sup> ð Þ *<sup>n</sup>*þ<sup>1</sup> *<sup>=</sup>*<sup>4</sup>

*e* � �*c*<sup>1</sup> � �*<sup>n</sup>*

> ð Þ 2*c*2*n* ! <sup>2</sup>*c*2*n*ð Þ *<sup>c</sup>*2*<sup>n</sup>* !

> > ð Þ 2*c*2*n* ! <sup>2</sup>*c*2*n*ð Þ *<sup>c</sup>*2*<sup>n</sup>* !

ð Þ 2*c*2*n* ! <sup>2</sup>*c*2*n*ð Þ *<sup>c</sup>*2*<sup>n</sup>* !

*,* (51)

(52)

(53)

(54)

(55)

(56)

(58)

*,* (57)

*,* (59)

<sup>2</sup>*π<sup>n</sup>* <sup>p</sup> *<sup>n</sup> e* � �*<sup>n</sup>*

where *<sup>m</sup>* <sup>¼</sup> <sup>Δ</sup>*<sup>n</sup>* 2 .

For the details of the proof of Theorems 1. 9 and 1.10, see [28, 29], respectively.
