end

By applying this operation *n* times, the coefficient of *xk*

From (27), we obtain the following relation

*Cp* <sup>¼</sup> <sup>X</sup> *p*�1

*k*¼1

*p* 2 � �

� 1 *p* X *p*�1

number of connected graphs *Cp* of orders going from *p* ¼ 1 to *p* ¼ 20.

% recurrence satisfied by the number of connected graphs

% C\_p= 2ˆ{combinations(p,2)}-1/p\* \sum\_{k=1}ˆ{p-1} k\* % combinations(p,k)\*2ˆ{combinations(p-k,2)}\*C\_{k}

*k*¼1

*<sup>k</sup> <sup>p</sup> k* � � 2

From (30), the following MATLAB code can be implemented for calculating the

acum = acum + j \* combinaciones(k,j) \* CuentaGrafEtiq(k-j) \* C(1,j);

sprintf('end function z=combinaciones(n,k) z= factorial(n)/(factorial(k)

*p* � *k* 2 � �

trees by inversions, and then deduced

*Probability, Combinatorics and Control*

can be expressed mathematically as

*Cp* ¼ 2

function y = CuentaGrafConnEtiq( p )

% Harary Graph Enumeration pag. 7

C(1,k) = CuentaGrafEtiq(k)-(1/k)\*acum;

z= factorial(n)/(factorial(k)\*factorial(n-k));

C(1:p)=0; C(1,1)=1; for k=2:p acum=0; for j=1:k

end

\*factorial(n-k));

function z=combinaciones(n,k)

end y=C(1,p);

end

**230**

number of labeled graphs of order *k* is obtained with exact *n* components

*G x*ð Þ¼ <sup>X</sup><sup>∞</sup>

1 þ *G x*ð Þ¼ *e*

*p* � 2 *k* � 1 � �

Riordan in [12] obtained the relation *Cp* ¼ *Jp*ð Þ2 , where *Jp*ð Þ *x* enumerates the

From (29), it should be noted that if the exponential generating function for the graph class is known in advance, then the exponential generating function for the class of graphs will be the logarithm of the first series, just as in (28). It should also be mentioned that another equivalent recurrence function can be obtained for the enumeration of the connected graphs that have order tags *p* (p. 7 in [13]) and that

*n*¼1

*<sup>C</sup>n*ð Þ *<sup>x</sup>*

*<sup>k</sup>*! that corresponds to the

*<sup>n</sup>*! *:* (27)

*C x*ð Þ*:* (28)

<sup>2</sup>*<sup>k</sup>* � <sup>1</sup> � �*CkCp*�*<sup>k</sup>:* (29)

*Ck:* (30)

%% calling the function from the matlab prompt for the calculation of the %% evaluation from graphs of order 1 to 20

>> for i=1:20 R(1,i)=CuentaGrafConnEtiq( i ); end

The calculations obtained by the execution of MATLAB code are shown in **Table 1**.

From the results of 1, it can be observed that the number of possible connected graphs *Cp* grows very fast in terms of the number *p* of vertices. It should be mentioned that (29) and (30) are recurrence relations instead of a closed formula. The recurrences (29) or (30) can be used for the calculation of *Cp* with a computer program. The generating functions can be used to solve recurrences and obtain a closed mathematical expression for the *n*th term of the succession associated with the recurrence. It can happen that calculation of the solution of some recurrences becomes very hard to be solved and in the worst cannot be solved at all. An alternative method for obtaining an approximate value for big values of *p* is to recur to the application of methods used in analytic combinatorics and calculate accurate approximations of the *p*th coefficient of the generating function. The generating functions algebraic structure allows to reflect the structure of combinatorial classes.


**Table 1.** *Order 1–20.* The analytic combinatorics method consists in examining the generating functions from the point of view of the mathematical analysis by giving not only real value values to its variables but also values in the complex plane. When complex values are assigned to the variables of the generating functions, the function is converted in a geometric transformation of the complex plane. This kind of geometrical mapping is said to be regular (holomorphic) near the origin of the complex plane. When we move away from the origin of the complex plane, some singularities appear that are related with the absence of smoothness of the function and give a lot of information about the function coefficients and their asymptotic growth. It can happen that elementary real analysis is enough for estimating asymptotically enumerative successions. If this is not the case, the generating functions are still explicit, but its form does not allow the easy calculation of the coefficients of the series. The complex plane analysis however is a good option for asymptotic estimation of these coefficients. In order to give an example of the use of the notion of singularities, let us take the ordinary generating function of the Catalan numbers

$$f(\mathbf{x}) = \frac{1}{2} \left( \mathbf{1} - \sqrt{\mathbf{1} - 4\mathbf{x}} \right). \tag{31}$$

We can evaluate in the same manner (31) giving to *x* values in the complex plane whose modulus is less than the radius of convergence of the series defined by (31) and realize that the orthogonal and regular grid in it transforms the real plane in a grid on the complex plane that preserves the angles of the curves of the grid. This property corresponds to the complex differentiability property, which also is equivalent to the property of analyticity. Concerning the asymptotic behavior of the coefficients *fn* of the generating function, it should be observed that it has a general

tial growth factor can be put in relation with the radius of convergence of the series

the complex plane that normally corresponds to the pole of the generating function,

2

The exponential growth part of (37) is known as first principle of coefficient asymptotics and the subexponential growth part as second principle of coefficient asymptotics. By recalling to the results that can be found in the field complex variable theory, more general generating functions can be obtained. One of those results is the Cauchy residue theorem that relates global properties of a meromorphic function (its integral along closed curves) to purely local characteristics at the residues poles. An important application of the Cauchy residue theorem concerns a

coefficient of analytic functions. This is stated in the following theorem [3]: Theorem 1.5 (Cauchy's coefficient formula). Let *f z*ð Þ be analytic in a region containing 0, and let *λ* be a simple loop around 0 that is positively oriented. Then,

*<sup>f</sup> <sup>n</sup>* � *<sup>z</sup><sup>n</sup>* ½ � *f z*ð Þ¼ <sup>1</sup>

2*iπ* ð *λ*

For more details about analytic combinatorics, we recommend to consult [3] as

A graph is a structure composed by a set of vertices *V* ¼ f g *v*1*; v*1*;* …*vn* and a set of

pairs of connected vertices *E* ¼ f g *e*1*;e*2*;* …*;em* called edges where *E*⊂*V* � *V* and

vertices, the number of possible trees that can be built is equal to ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *<sup>n</sup>*�<sup>1</sup>

[10] gives an example for the case of four vertices 4 ¼ *n* þ 1 then *n* ¼ 3, and the total number of possible trees calculated using his formula gives 4ð Þ<sup>2</sup> <sup>¼</sup> 16. In this

of graph that do not have cycles. A tree have no loops or edges that connect a vertex with himself. The subject of combinatorial graph enumeration has been the center of interest of many mathematicians a long time ago. The enumeration of total possible labeled trees with *n* nodes being *nn*�<sup>2</sup> was one of first results obtained by Cayley in [10]. Cayley's formula for enumerating trees is one of the simple and elegant mathematical results in enumeration of graphs. He detected that from *n* þ 1

*f z*ð Þ *dz*

*zn*þ<sup>1</sup> *:* (38)

� �. A tree is a special type

. Cayley

the coefficient *<sup>z</sup><sup>n</sup>* ½ � *f z*ð Þ admits the integral representation:

each edge *ek* ∈ *E* is composed as pair of vertices *ek* ¼ *vi; vj*

that is the singularity that can be observed along the positive real axis of

4

*<sup>x</sup><sup>n</sup>* ½ � *f x*ð Þ¼ *Anθ*ð Þ *<sup>n</sup> :* (37)

� � arises from the singularity of the square

ffiffiffiffiffiffiffi *πn*<sup>3</sup> � � p �<sup>1</sup>

, the exponen-

asymptotic pattern composed by an exponential growth factor *An* and a

For the case of the expression (34) *<sup>A</sup>* <sup>¼</sup> 4 and *<sup>θ</sup>*ð Þ� *<sup>n</sup>* <sup>1</sup>

root type. This asymptotic behavior can be compactly expressed as

subexponential factor *θ*ð Þ *n* .

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

and the subexponential part *<sup>θ</sup>*ð Þ¼ *<sup>n</sup> O n*�<sup>3</sup>

by *<sup>A</sup>* <sup>¼</sup> <sup>1</sup> *ρf*

well as [5].

**233**

**4. Enumeration of trees**

Eq. (31) expresses in a compact way the power series of the form

$$\left(\left(1-y\right)^{1/2} = 1 - \frac{1}{2}y - \frac{1}{8}y^2 - \dots \tag{32}$$

The generating function (31) coefficients can be explicitly expressed as

$$f\_n = [\mathbf{x}^n] f(\mathbf{x}) = \frac{1}{n} \binom{2n-2}{n-1}.\tag{33}$$

Using the Stirling formula, we can get the asymptotic approximation of (33) that is expressed as

$$f\_n \sim \lim\_{n \to \infty} \frac{4^n}{\sqrt{\pi n^3}}.\tag{34}$$

If the generating function is used as an analytic object, the approximation (34) can be obtained.

In order to do it, we substitute in the power series expansion of the generating function *f*(*x*) any real or complex value *ρ<sup>f</sup>* whose modulus is small enough, for example, *ρ<sup>f</sup>* ¼ 4. The graph that we get by the use of (31) is smooth and differentiable in the real plane and tends to the limit <sup>1</sup> <sup>2</sup> as *<sup>x</sup>* ! <sup>1</sup> 4 � ��, but, if we calculate its derivative, we obtain the following function

$$f(\mathbf{x}) = \frac{1}{1 - \sqrt{1 - 4\mathbf{x}}},\tag{35}$$

and it can be noticed that the derivative (35) becomes infinite in *<sup>ρ</sup><sup>f</sup>* <sup>¼</sup> <sup>1</sup> 4. The singularities will correspond to those points where the graph is not smooth.

It should be pointed out that that the region where function (31) is still being continuous can be extended. Let us take for example the value *x* ¼ �1

$$f(-1) = \frac{1}{2} \left( 1 - \sqrt{5} \right). \tag{36}$$

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

The analytic combinatorics method consists in examining the generating functions from the point of view of the mathematical analysis by giving not only real value values to its variables but also values in the complex plane. When complex values are assigned to the variables of the generating functions, the function is converted in a geometric transformation of the complex plane. This kind of geometrical mapping is said to be regular (holomorphic) near the origin of the complex plane. When we move away from the origin of the complex plane, some singularities appear that are related with the absence of smoothness of the function and give a lot of information about the function coefficients and their asymptotic growth. It can happen that elementary real analysis is enough for estimating asymptotically enumerative successions. If this is not the case, the generating functions are still explicit, but its form does not allow the easy calculation of the coefficients of the series. The complex plane analysis however is a good option for asymptotic estimation of these coefficients. In order to give an example of the use of the notion of singularities, let us take the ordinary generating function of the Catalan numbers

*Probability, Combinatorics and Control*

*f x*ð Þ¼ <sup>1</sup> 2

ð Þ 1 � *y*

is expressed as

can be obtained.

**232**

tiable in the real plane and tends to the limit <sup>1</sup>

derivative, we obtain the following function

Eq. (31) expresses in a compact way the power series of the form

*<sup>f</sup> <sup>n</sup>* <sup>¼</sup> *xn* ½ � *f x*ð Þ¼ <sup>1</sup>

<sup>1</sup>*=*<sup>2</sup> <sup>¼</sup> <sup>1</sup> � <sup>1</sup>

The generating function (31) coefficients can be explicitly expressed as

*<sup>f</sup> <sup>n</sup>* � lim*<sup>n</sup>*!<sup>∞</sup>

*f x*ð Þ¼ <sup>1</sup>

and it can be noticed that the derivative (35) becomes infinite in *<sup>ρ</sup><sup>f</sup>* <sup>¼</sup> <sup>1</sup>

It should be pointed out that that the region where function (31) is still being

1 2

<sup>1</sup> � ffiffi 5

singularities will correspond to those points where the graph is not smooth.

continuous can be extended. Let us take for example the value *x* ¼ �1

*f*ð Þ¼ �1

2 *<sup>y</sup>* � <sup>1</sup>

*n*

Using the Stirling formula, we can get the asymptotic approximation of (33) that

If the generating function is used as an analytic object, the approximation (34)

In order to do it, we substitute in the power series expansion of the generating function *f*(*x*) any real or complex value *ρ<sup>f</sup>* whose modulus is small enough, for example, *ρ<sup>f</sup>* ¼ 4. The graph that we get by the use of (31) is smooth and differen-

<sup>1</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffi

4*n* ffiffiffiffiffiffiffi

2*n* � 2 *n* � 1 � �

<sup>2</sup> as *<sup>x</sup>* ! <sup>1</sup>

4

<sup>1</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � 4*x*

� � <sup>p</sup> *:* (31)

<sup>8</sup> *<sup>y</sup>*<sup>2</sup> � …*:* (32)

*<sup>π</sup>n*<sup>3</sup> <sup>p</sup> *:* (34)

<sup>1</sup> � <sup>4</sup>*<sup>x</sup>* <sup>p</sup> *,* (35)

� � <sup>p</sup> *:* (36)

*:* (33)

� ��, but, if we calculate its

4. The

We can evaluate in the same manner (31) giving to *x* values in the complex plane whose modulus is less than the radius of convergence of the series defined by (31) and realize that the orthogonal and regular grid in it transforms the real plane in a grid on the complex plane that preserves the angles of the curves of the grid. This property corresponds to the complex differentiability property, which also is equivalent to the property of analyticity. Concerning the asymptotic behavior of the coefficients *fn* of the generating function, it should be observed that it has a general asymptotic pattern composed by an exponential growth factor *An* and a subexponential factor *θ*ð Þ *n* .

For the case of the expression (34) *<sup>A</sup>* <sup>¼</sup> 4 and *<sup>θ</sup>*ð Þ� *<sup>n</sup>* <sup>1</sup> 4 ffiffiffiffiffiffiffi *πn*<sup>3</sup> � � p �<sup>1</sup> , the exponential growth factor can be put in relation with the radius of convergence of the series by *<sup>A</sup>* <sup>¼</sup> <sup>1</sup> *ρf* that is the singularity that can be observed along the positive real axis of the complex plane that normally corresponds to the pole of the generating function, and the subexponential part *<sup>θ</sup>*ð Þ¼ *<sup>n</sup> O n*�<sup>3</sup> 2 � � arises from the singularity of the square root type. This asymptotic behavior can be compactly expressed as

$$\left[\mathfrak{x}^{n}\right]f(\mathfrak{x}) = A^{n}\theta(n). \tag{37}$$

The exponential growth part of (37) is known as first principle of coefficient asymptotics and the subexponential growth part as second principle of coefficient asymptotics. By recalling to the results that can be found in the field complex variable theory, more general generating functions can be obtained. One of those results is the Cauchy residue theorem that relates global properties of a meromorphic function (its integral along closed curves) to purely local characteristics at the residues poles. An important application of the Cauchy residue theorem concerns a coefficient of analytic functions. This is stated in the following theorem [3]:

Theorem 1.5 (Cauchy's coefficient formula). Let *f z*ð Þ be analytic in a region containing 0, and let *λ* be a simple loop around 0 that is positively oriented. Then, the coefficient *<sup>z</sup><sup>n</sup>* ½ � *f z*ð Þ admits the integral representation:

$$\int f\_n \equiv [z^n] \, f(z) = \frac{1}{2i\pi} \int\_{\lambda} f(z) \frac{dz}{z^{n+1}}.\tag{38}$$

For more details about analytic combinatorics, we recommend to consult [3] as well as [5].

## **4. Enumeration of trees**

A graph is a structure composed by a set of vertices *V* ¼ f g *v*1*; v*1*;* …*vn* and a set of pairs of connected vertices *E* ¼ f g *e*1*;e*2*;* …*;em* called edges where *E*⊂*V* � *V* and each edge *ek* ∈ *E* is composed as pair of vertices *ek* ¼ *vi; vj* � �. A tree is a special type of graph that do not have cycles. A tree have no loops or edges that connect a vertex with himself. The subject of combinatorial graph enumeration has been the center of interest of many mathematicians a long time ago. The enumeration of total possible labeled trees with *n* nodes being *nn*�<sup>2</sup> was one of first results obtained by Cayley in [10]. Cayley's formula for enumerating trees is one of the simple and elegant mathematical results in enumeration of graphs. He detected that from *n* þ 1 vertices, the number of possible trees that can be built is equal to ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *<sup>n</sup>*�<sup>1</sup> . Cayley [10] gives an example for the case of four vertices 4 ¼ *n* þ 1 then *n* ¼ 3, and the total number of possible trees calculated using his formula gives 4ð Þ<sup>2</sup> <sup>¼</sup> 16. In this

same publication, Cayley gives another example having *n* þ 1 ¼ 6 vertices (he used the term knots in his publication) with labels *a, b, c, d, e, f* and related the concatenation of vertices given by the edges for obtaining sequences of labels representing a given tree. For instance, if the tree is a chain of vertices connected by edges starting with the vertex *a* and ending with the vertex *f* and having as connecting edges ð Þ *a; b , b*ð Þ *;c , c*ð Þ *; d , d*ð Þ *;e , e*ð Þ *; f* , the corresponding label sequence is *abcdef* as shown in **Figure 1**. As another example of sequence of vertex labels, if the root of the tree is *α* and it is connected directly with the other five vertices, then the connecting edges will be ð Þ *a; b , a*ð Þ *;c , a*ð Þ *; d , a*ð Þ *;e , a*ð Þ *; f* , and the corresponding sequence of vertex labels will be in that case *a*5*bcdef* as shown in **Figure 2**. As can be noticed, the exponent of *a* is 5 that represents the number of occurrences of this label in the set of connecting edges.

After that Cayley states the theorem for this particular case as follows:

Theorem 1.6 The total number of trees *T n*ð Þ þ 1 that can be built with *n* þ 1 ¼ 6 vertices can be calculated as follows

$$T(n+1) = (a+b+c+d+e+f)^4 \\ abcdef = 6^4 = 1296 \tag{39}$$

**5. Enumeration and generating functions**

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

numbers whose *n*th element is *an* ¼ 2*n* � 1.

two types of graph enumerating problems:

1.Labeled graph problems

**235**

2.Unlabeled graph problems

series.

The field of combinatorial enumeration has aroused enormous interest among mathematicians who have worked in the area of discrete mathematics during the last decades [18–21]. Combinatorial enumerative technique developed by these brilliant researchers have allowed us to count words, permutations,

partitions, sequences, and graphs. As was mentioned in Section 3, the mathematical tool frequently used for this purpose is the generating functions or formal power

The generating functions allow to connect discrete mathematics and continuous analysis in a very special way with complex variable theory. The typical situation that someone faces when trying to solve an enumeration problem is that you want to know the mathematically closed form that has the *n*th term of a given sequence of numbers *a*0*, a*1*, a*2*,* … For some sequences, we can do by inspection. For example, if the numerical sequence 1*;* 3*;* 5*;* 7*;* 9*,* …, it is easy to see that it is a sequence of odd

A more complicated sequence is the set of prime numbers 2*;* 3*;* 5*;* 7*;* 11*;* 13*;* 17*;* 19*,* …,

In many cases it is very hard to get a simple formula just by inspection. However

*aix<sup>i</sup>*

Eq. (40) defines an ordinary generating function. As mentioned in Section 3, since they are infinite polynomials, they can be algebraically manipulated easily. In this chapter, the main interest will be the application of the generating functions tool as well as the analytic asymptotic methods for the enumeration of graphs accomplishing some given properties. Many questions about the number of graphs that have some specified property can be answered by the use of generating functions. Some typical questions about the number of graphs that fulfill a given property are, for example: How many different graphs can I build with *n* vertices? How many different connected graphs with *n* vertices exist? How many binary trees can be constructed with *n* vertices? [18, 19], etc. For some of these questions, the application of generating functions allows us to easily obtain a simple formula. For some other questions, the answer is an asymptotic estimation formula. The most commonly used generating functions are the *ordinary generating functions* and the *exponential generating functions*. The generating functions are the tool used for enumerating graphs. From the point of view of the generating functions, there are

The labeled graph problems can be easily solved with the direct application of the exponential generating functions. The case of the unlabeled enumeration problems can be solved by using ordinary generating functions but require the use of

more combinatorial theory and the application of Pólya's theorem.

*:* (40)

whose *an* is the *n*th prime number. A closed mathematical formula for *n*th prime

it can be very useful to use the generating functions whose coefficients are the

X∞ *i*¼0

number is not known, and it seems impossible to obtain in general.

elements of that sequence transforming it as follows:

This calculation relates the sum of the products of the coefficients of the multinomial ð Þ *<sup>a</sup>* <sup>þ</sup> *<sup>b</sup>* <sup>þ</sup> *<sup>c</sup>* <sup>þ</sup> *<sup>d</sup>* <sup>þ</sup> *<sup>e</sup>* <sup>þ</sup> *<sup>f</sup>* <sup>4</sup> with the number of terms of its corresponding type. Each term obtained by multiplying *abcdef* with the vertex label inside ð Þ *<sup>a</sup>* <sup>þ</sup> *<sup>b</sup>* <sup>þ</sup> *<sup>c</sup>* <sup>þ</sup> *<sup>d</sup>* <sup>þ</sup> *<sup>e</sup>* <sup>þ</sup> *<sup>f</sup>* <sup>4</sup> corresponds to different trees.

At the end of [10], Cayley generalizes his theorem by recalling a result obtained by C.W. Borchardt in [14] that relates some particular kind of determinants that represent spanning trees and whose product represents the branches of those spanning trees. Given that the number of terms of these determinants is ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *<sup>n</sup>*�<sup>1</sup> , Cayley can conclude that the number of spanning trees is the same. Since these first results, many other methods have been proposed for obtaining the same result. One way to enumerate a collection of objects is to find a bijection between a set of objects whose enumeration is known and the set of objects that we want to enumerate. This was the method used by Prüfer in [15] for enumerating the set of possible spanning trees with *n* vertices. The set whose number was known beforehand was a sequence of length *n* � 2 of numbers from 1 to *n*. For this end the autor of article [15] encoded the trees as Prüfer sequences. In [16] Moon generalized the result derived by Clarke in [17] by induction on *d* the degree, making induction on *n*, and the number of vertices and obtains a new enumeration method for *n*-labeled *k* trees.

**Figure 1.** *Example of tree of one branch.*

**Figure 2.** *Example of tree with more branches.*

same publication, Cayley gives another example having *n* þ 1 ¼ 6 vertices (he used the term knots in his publication) with labels *a, b, c, d, e, f* and related the concatenation of vertices given by the edges for obtaining sequences of labels representing a given tree. For instance, if the tree is a chain of vertices connected by edges starting with the vertex *a* and ending with the vertex *f* and having as connecting edges ð Þ *a; b , b*ð Þ *;c , c*ð Þ *; d , d*ð Þ *;e , e*ð Þ *; f* , the corresponding label sequence is *abcdef* as shown in **Figure 1**. As another example of sequence of vertex labels, if the root of the tree is *α* and it is connected directly with the other five vertices, then the connecting edges will be ð Þ *a; b , a*ð Þ *;c , a*ð Þ *; d , a*ð Þ *;e , a*ð Þ *; f* , and the corresponding sequence of vertex labels will be in that case *a*5*bcdef* as shown in **Figure 2**. As can be noticed, the exponent of *a* is 5 that represents the number of occurrences of this

After that Cayley states the theorem for this particular case as follows:

*T n*ð Þ¼ <sup>þ</sup> <sup>1</sup> ð Þ *<sup>a</sup>* <sup>þ</sup> *<sup>b</sup>* <sup>þ</sup> *<sup>c</sup>* <sup>þ</sup> *<sup>d</sup>* <sup>þ</sup> *<sup>e</sup>* <sup>þ</sup> *<sup>f</sup>* <sup>4</sup>

ð Þ *<sup>a</sup>* <sup>þ</sup> *<sup>b</sup>* <sup>þ</sup> *<sup>c</sup>* <sup>þ</sup> *<sup>d</sup>* <sup>þ</sup> *<sup>e</sup>* <sup>þ</sup> *<sup>f</sup>* <sup>4</sup> corresponds to different trees.

Each term obtained by multiplying *abcdef* with the vertex label inside

Theorem 1.6 The total number of trees *T n*ð Þ þ 1 that can be built with *n* þ 1 ¼ 6

This calculation relates the sum of the products of the coefficients of the multinomial ð Þ *<sup>a</sup>* <sup>þ</sup> *<sup>b</sup>* <sup>þ</sup> *<sup>c</sup>* <sup>þ</sup> *<sup>d</sup>* <sup>þ</sup> *<sup>e</sup>* <sup>þ</sup> *<sup>f</sup>* <sup>4</sup> with the number of terms of its corresponding type.

At the end of [10], Cayley generalizes his theorem by recalling a result obtained by C.W. Borchardt in [14] that relates some particular kind of determinants that represent spanning trees and whose product represents the branches of those spanning trees. Given that the number of terms of these determinants is ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *<sup>n</sup>*�<sup>1</sup>

Cayley can conclude that the number of spanning trees is the same. Since these first results, many other methods have been proposed for obtaining the same result. One way to enumerate a collection of objects is to find a bijection between a set of objects whose enumeration is known and the set of objects that we want to enumerate. This was the method used by Prüfer in [15] for enumerating the set of possible spanning trees with *n* vertices. The set whose number was known beforehand was a sequence of length *n* � 2 of numbers from 1 to *n*. For this end the autor of article [15] encoded the trees as Prüfer sequences. In [16] Moon generalized the result derived by Clarke in [17] by induction on *d* the degree, making induction on *n*, and the number of vertices and obtains a new enumeration method for *n*-labeled

*abcdef* <sup>¼</sup> 64 <sup>¼</sup> 1296 (39)

,

label in the set of connecting edges.

*Probability, Combinatorics and Control*

vertices can be calculated as follows

*k* trees.

**Figure 1.**

**Figure 2.**

**234**

*Example of tree of one branch.*

*Example of tree with more branches.*
