Journal of Applied Mathematics

Volume 2014 (2014), Article ID 428020, 6 pages

http://dx.doi.org/10.1155/2014/428020

Research Article

On -Distance Pell Numbers in 3-Edge-Coloured Graphs

Krzysztof Piejko and Iwona Włoch

Faculty of Mathematics and Applied Physics, Rzeszow University of Technology, Aleja Powstańców Warszawy 12, 35-959 Rzeszów, Poland

Received 27 May 2014; Revised 17 July 2014; Accepted 19 July 2014; Published 5 August 2014

Academic Editor: Song Cen

Copyright © 2014 Krzysztof Piejko and Iwona Włoch. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We define in this paper new distance generalizations of the Pell numbers and the companion Pell numbers. We give a graph interpretation of these numbers with respect to a special 3-edge colouring of the graph.

1. Introduction

The Fibonacci sequence is defined by the following recurrence relation for with . Among sequences of the Fibonacci type there is the Pell sequence defined by for with the initial conditions and . The companion Pell sequence is closely related to the Pell sequence and is given by formula and for . The Pell sequences play an important role in the number theory and they have many interesting interpretations. We recall some of them.

(i)The number of lattice paths from the point to the line consisting of , , and steps is equal to ; see [1].(ii)The number of compositions (i.e., ordered partitions) of a number into two sorts of of 1’s and one sort of 2’s is equal to , see [1].(iii)The number of -step non-self-intersecting paths starting at the point with steps of types , , or is equal to , see [1].

Interesting generalizations of the numbers of the Fibonacci type (also generalizations of the Pell sequences) are studied, for instance, by Kilic in [2–4]. Among others Kilic in [4] introduced the generalization of the Pell numbers. He defined the generalized Pell -numbers for any given , where and for and in the following way with initial conditions and ,  . If then .

In this paper we describe new kinds of generalized Pell sequence and the companion Pell sequence. Our generalization is closely related to the recurrence given in [4] by Kilic. By other initial conditions we obtain other generalized Pell sequences. We give their graph interpretations which are closely related to a concept of edge colouring in graphs. Graph interpretations of the Fibonacci numbers and the like are study intensively; see, for example, [5–9].

2. -Distance Pell Sequences and

Let , be integers. The -distance Pell sequence is defined by the th order linear recurrence relation:

with the following initial conditions:

If , then this definition reduces to the classical Pell numbers; that is, .

Table 1 includes a few first words of the for special values of .

Table 1: The -distance Pell sequence .

Firstly we give an interpretation of the -distance Pell sequence, which generalizes result given in (i).

Theorem 1. Let and be integers. Then the number of lattice paths from the point to the line consisting of , , and steps is equal to .

Proof. Let be the number of paths from the point to the line . It is easy to notice that for .

Let and let , , and denote the number of such paths from the point to the line , for which the last step is of the form , , and , respectively.

It can be easily seen that , , and . Since then we have and consequently , for all .

In the same way we can prove the generalization of the result (ii).

Theorem 2. Let and be integers. Then the number of all compositions of the number into two sorts of 1’s and one sort of ’s is equal to .

By analogy to the Pell sequence we introduce a generalization of the companion Pell sequence which generalizes the classical companion Pell sequence in the distance sense.

Let , be integers. The -distance companion Pell sequence is defined by the th order linear recurrence relation:

with the initial conditions

If then gives the classical companion Pell numbers ; that is, .

Table 2 includes a few first words of the for special values of .

Table 2: The -distance companion Pell sequence .

The following theorem gives the basic relation between and .

Theorem 3. Let and be integers. Then

Proof (by induction on ). For the result follows immediately by the definitions of and . Let . Assume that formula (5) is true for , . We will prove that .

By the induction hypothesis and the definitions of and , we have that

which ends the proof.

For Theorem 3 gives the well-known relation between the classical Pell numbers and the companion Pell numbers; namely, .

Theorem 4. Let , , and be integers. Then

Proof of (7) (by induction on ). If then the equation is obvious. Let . Assume that formula (7) is true for an arbitrary . We will prove that .

By the induction hypothesis and the definition of we have

which ends the proof of (7).

In the same way we can prove the equality (8), so we omit the proof.

If or , respectively, then from Theorem 4 the following follows.

Corollary 5. Let and be integers. Then

For we obtain the well-known identities for the classical Pell numbers and the companion Pell numbers; namely,

Theorem 6. Let and . Then

Proof of (12) (by induction on ). If then the result is obvious. Assume that for an arbitrary , such that . We will prove that .

By the induction hypothesis and the definition of , we have

which ends the proof.

In the same way we can prove the equality (13).

If , , or , respectively, then from Theorem 6 we obtain the following corollary.

Corollary 7. Let . Then

Theorem 8. Let and be integers. Then

Proof of (16) (by induction on ). For we have the equation

By the initial conditions of the sequence and Theorem 6 we can see that this equation is an identity (because it is equivalent to third identity from Corollary 7).

Let . Assume that formula (16) is true for an arbitrary where . We will prove that . By the induction hypothesis and the definition of , we have

which ends the proof of (16). In the same way we can prove (17) which completes the proof of the theorem.

If then from Theorem 8 we obtain the following formula for the classical Pell numbers and companion Pell numbers:

3. -Coloured Graphs

For concepts not defined here see [10]. The numbers of the Fibonacci type have many applications in distinct areas of mathematics. There is a large interest of modern science in the applications of the numbers of the Fibonacci type. These numbers are studied intensively in a wide sense also in graphs and combinatorials problem. In graphs Prodinger and Tichy initiated studying the Fibonacci numbers and the like. In [11] they showed the relations between the number of independent sets in and with the Fibonacci numbers and the Lucas numbers, where and denote an -vertex path and an -vertex cycle, respectively. This short paper gave an impetus for counting problems related to the numbers of the Fibonacci type. Many of these problems and results are closely related with the Merrifield-Simmons index and the Hosoya index in graphs; see [6, 12]. The Pell numbers also have a graph interpretation. It is well-known that , where denotes the corona of two graphs.

In this section we give a graph interpretation of the -distance Pell numbers with respect to special edge colouring of a graph.

Let be a 3-edge coloured graph with the set of colours . Let . We say that a path is -monochromatic if all its edges are coloured alike by colour . By we denote the length of the -monochromatic path. For notation means that the edge has the colour .

Let be an integer. In the graph we define a -edge colouring, such that , , and , where is an integer.

Theorem 9. Let and be integers. The number of all -edge colouring of the graph is equal to .

Proof. Let be the set of vertices of a graph with the numbering in the natural fashion. Then . Let be the number of all -edge colouring of the graph . By inspection we obtain that for .

Let and let , , and denote the number of -edge colouring of the graph , with , , and , respectively. It is obvious that .

Clearly and are equal to the number of all -edge colouring of the graph and is equal to the number of all -edge colouring of the graph . In the other words , , and . Since

then we have . By the initial conditions we have that , for all , which ends the proof.

Corollary 10. Let be integer. The number of all -edge colouring of the graph is equal to .

Using the concept of -edge colouring of the graph we can obtain the direct formula for the numbers and .

Let , , and be integers and let be the number of -edge colouring of graph , such that -monochromatic path appears in this colouring exactly times. In the other words edges of have colour .

Theorem 11. Let and be integers. Then .

Proof. Since has edges, then for we obtain that

and so .

Theorem 12. Let , , and be integers. Then for all holds .

Proof. Consider the graph with the -edge colouring. By contracting the -monochromatic paths of length to the -monochromatic paths of length we can see that the number of -edge colouring of graph , such that -monochromatic path appears in this colouring exactly times is equal to the number of -edge colouring of graph , such that -monochromatic path appears in this colouring exactly times and the Theorem follows.

If or then Theorem 12 gives the following result.

Corollary 13. Let , , and be integers. Then

By (23) and Theorem 11 we obtain the direct formula for .

Corollary 14. Let , , be integers. Then

Moreover Theorem 9 immediately gives that

Using (26) and Corollary 14 we can give the direct formula for the number .

Theorem 15. Let and be integers. Then

If then we obtain the direct formula for the classical Pell numbers of the form

From Theorems 3 and 15 follows the direct formula for the sequence .

Theorem 16. Let and be integers. Then

If then using the above theorem and after some calculations we obtain that

Now we give the recurrence relation for the number .

Theorem 17. Let , , and be integers. Then

Proof. Let be the set of vertices of a graph with the numbering in the natural fashion. Then .

Let , , and denote the number of -edge colouring of the graph , such that -monochromatic path appears in this colouring exactly times with , , and , respectively.

It can be easily seen that both and are equal to the number of all -edge colouring of the graph , such that -monochromatic path appears in this colouring exactly times and is equal to the number of all -edge colouring of the graph , such that -monochromatic path appears in this colouring exactly times. In the other words , , and . Since

so the result immediately follows.

By (24) and by Theorem 17 we obtain the following.

Corollary 18. Let , , and be integers. Then

Using (26) and Corollary 18 we have the following.

Corollary 19. Let , , and be integers. Then

Now we consider the -edge colouring of the cycle with the numbering of edges in the natural fashion. For explanation for cycle with there are two distinct -edge colouring using only colour . The first -edge colouring gives a partition of the set and the second -edge colouring gives a partition .

Let be the number of all -edge colouring of the cycle .

Theorem 20. Let and be integers. The number of all -edge colouring of the cycle is equal to .

Proof. Let be the set of vertices of a graph with the numbering in the natural fashion. Then .

It is easy to check that for .

Let and let , , and denote the number of -edge colouring of the cycle with , , and , respectively.

It can be easily seen that and are equal to the number of all -edge colouring of the graph and is equal to the number of all -edge colouring of the graph multiplied by . By Theorem 9 we have , , and . Since

then we have and so by Theorem 3 we have for all .

Corollary 21. Let be an integer. Then the number of all -edge colouring of the cycle is equal to , .

4. Concluding Remarks

The interpretation of the generalized Pell numbers with respect to -colouring of a 3-edge coloured graph gives a motivation for studying this type of colouring in graphs. For an arbitrary this problem seems to be difficult and more interesting results can be obtained for special value of (e.g., if we study -colouring in graphs). In the class of trees some interesting results can be obtained.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the referee for helpful valuable suggestions which resulted in improvements to this paper.

References

The On-Line Encyclopedia of Integer Sequences

E. Kilic, “On the usual Fibonacci and generalized order- Pell numbers,” Ars Combinatoria, vol. 88, pp. 33–45, 2008.

E. Kilic and D. Tasci, “On the generalized Fibonacci and Pell sequences by Hessenberg matrices,” Ars Combinatoria, vol. 94, pp. 161–174, 2010.

E. Kilic, “The generalized Pell -numbers and their Binet formulas, combinatorial representations, sums,” Chaos, Solitons & Fractals, vol. 40, no. 4, pp. 2047–2063, 2009.

D. Bród, K. Piejko, and I. Włoch, “Distance Fibonacci numbers, distance Lucas numbers and their applications,” Ars Combinatoria, vol. 112, pp. 397–409, 2013.

H. Hosoya, “Topological index. A nawly proposed quantity chracterizing the topological nature of structural isomers of saturated hydrocarbons,” Bulletin of the Chemical Society of Japan, vol. 44, no. 9, pp. 2332–2339, 1971.

I. Włoch and A. Włoch, “Generalized sequences and -independent sets in graphs,” Discrete Applied Mathematics, vol. 158, no. 17, pp. 1966–1970, 2010.

A. Włoch and M. Wołowiec-Musiał, “Generalized Pell numbers and some relations with Fibonacci numbers,” Ars Combinatoria, vol. 109, pp. 391–403, 2013.

A. Wloch and I. Wloch, “On generalized pell numbers and their graph representations,” Commentationes Mathematicae, vol. 48, no. 2, pp. 169–175, 2008.

R. Diestel, Graph Theory, Springer, New York, NY, USA, 2005.

H. Prodinger and R. F. Tichy, “Fibonacci numbers of graphs,” The Fibonacci Quarterly, vol. 20, no. 1, pp. 16–21, 1982.

S. Wagner and I. Gutman, “Maxima and minima of the Hosoya index and the Merrifield-Simmons index: a survey of results and techniques,” Acta Applicandae Mathematicae, vol. 112, no. 3, pp. 323–346, 2010.