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Vol. 19 (2018), No. 2, pp. 741–748 DOI: 10.18514/MMN.2018.1415

LINK BETWEEN HOSOYA INDEX AND FIBONACCI NUMBERS

HAC `ENE BELBACHIR AND HAKIM HARIK Received 09 November, 2014

Abstract. LetGbe a graph andZ.G/be its Hosoya index. We show how the Hosoya index can be a good tool to establish some new identities involving Fibonacci numbers. This permits to extend Hillard and Windfeldt work.

2010Mathematics Subject Classification: 05A19; 05C30; 11B39 Keywords: Fibonacci numbers, matching, Hosoya index, paths

1. INTRODUCTION

We denote by G D.V .G/; E.G// a simple undirected graph, V .G/ is the set of its vertices andE.G/ is the set of its edges. The order ofG isjV .G/jand the size of G is jE.G/j. For a vertex v ofG; N.v/ is the set of vertices adjacent to v, deg.v/WD jN.v/jis the degree of vILi nk.v/is the set of edges incident to v.

An edge fu; vg ofG is denoted uv: A path Pn, from a vertex v1 to a vertex vn, n2, is a sequence of verticesv1; : : : ; vn and edges viviC1, for i D1; : : : ; n 1I for simplicity we denote it byv1 vn. We extend the definition ofPntonD0and nD1by settingP0 is empty andP1 is a single vertex, we add the convention that PnP0DP0PnDPnfor alln1.

The graphG vis obtained fromG by removing the vertexvand all edges ofG which are incident tov. For an edgeeofG;we denote byG ethe graph obtained fromG by removing e. The contraction of a graph G;associated to an edge e;is the graphG=e obtained by removingeand identifying the verticesuandvincident toe and replacing them by a single vertexv0where any edges incident touorv are redirected tov0. We then say that we contract inG the adjacent verticesu; vinto the vertexv0:

The well-known Fibonacci sequencefFngis defined asF0D0; F1D1;andFnD Fn 1CFn 2, for n2. The Fibonacci numbers are connected to the element of Pascal’s triangle using the following identity

FnC1D

n

X

kD0

n k

k

! :

c 2018 Miskolc University Press

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For some results and properties related to Fibonacci numbers, see for instance [1].

Many fields widely applies this sequence, particularly in physics and chemistry [10].

A matching M of a graphG is a subset of E.G/such that no two edges inM share a vertex inG:A matching ofG is also called an independent edge set ofG:A k-matching of a graphG is a matching ofG of cardinalk, it is then an independent edge set ofG of cardinalk:We denote bym .G; k/the number ofk-matchings ofG with the conventionm .G; 0/D1:Note thatm .G; 1/D jE.G/jand whenk > n=2, m .G; k/D0:

The Hosoya index of a graph G, denoted by Z.G/, is an index introduced by Hosoya [9] as follows.

Z .G/D

bn=2c

X

kD0

m .G; k/ ;

wherenD jV .G/j,bn=2cstands for the integer part ofn=2:This index has several applications in molecular chemistry such as boiling point, entropy or heat of vapor- ization. The literature includes many papers dealing the Hosoya index [2,3,6].

2. PRELIMINARY RESULTS

Before proving our main results, we first list the following results. From the defin- ition of the Hosoya index, it is not difficult to deduce the following Lemma.

Lemma 1([7]). LetGbe a graph, we have

.1/Ifuv2E .G/ ;thenZ .G/DZ .G uv/CZ .G fu; vg/ : .2/Ifv2V .G/ ;thenZ.G/DZ.G v/CP

w2NG.v/Z.G fw; vg/.

.3/IfG1; G2; :::; Gt are the components ofG thenZ .G/DQt

kD1Z .Gk/ : Lemma1allows us to computeZ.G/for any graph recursively.

The following theorem gives a relation between Hosoya index and Fibonacci num- ber (see [5], [7]).

Theorem 1. LetPnbe a path onnvertices, thenZ.Pn/DFnC1:

3. MAIN RESULTS

In this section, inspired by [4], we give another proofs of well-known identities (see Lemmas2and3). Our goal is to prove the formula of Lemma3 via Theorem 1. For this we give a direct proof of Lemma2. We also establish two new identities, given in Theorems2and3.

The following identity shows the relation between independent edge subsets in Pr1Cr2; Pr1 andPr2 forr1andr2two non-negative integers.

Lemma 2. Letr1; r2be two non-negative integers, then

Z .Pr1Cr2/DZ .Pr1/ Z .Pr2/CZ .Pr1 1/ Z .Pr2 1/ :

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Proof. Letv1 vr1Cr2be a pathPr1Cr2 partitioned into two pathsPr1 represen- ted by a sequence of vertices v1 vr1 and Pr2 represented by a sequence of ver- tices vr1C1 vr1Cr2: The vertices vr1 and vr1C1 share the same edge in Pr1Cr2: The Hosoya numberZ .Pr1Cr2/of the pathPr1Cr2 represents the number of inde- pendent edge subsets between the vertices of Pr1Cr2;this index can be written as Z .Pr1Cr2/D jMj C jM0j;where:

M is the set of independent edge subsets ofPr1Cr2such that the edgevr1vr1C1

does not belong to any independent edge subset ofM, that means all inde- pendent edge subsets ofM are inPr1 andPr2;sojMj DZ .Pr1/ Z .Pr2/.

M0 is the set of independent edge subsets of Pr1Cr2 such that the edge vr1vr1C1belongs to all independent edge subsets ofM0, that means the oth- ers independent edges of every subset of M0 are in Pr1 1 and Pr2 1; so jMj DZ .Pr1 1/ Z .Pr2 1/.

Lemma 3. Letk; nbe two integers such that1kn:Then

FnC1DFkFn kCFkC1Fn kC1:

Proof. Consider a pathPnDv1 vnonnvertices. We setr1WDk; r2WDn k;

and use Theorem1and Lemma2.

We introduce a new identity of Fibonacci numbers which generalize identities of Fibonacci numbers given in [8].

For every integers2, let˝sbe the set of"WD."1; "2; :::; "s/with"i 2 f 1; 0; 1g .1is/such that :

(1) The number of"i D0is even. Let2h."/this number.

(2) If2h."/D0, then"iD1for alli.

(3) If2h."/¤0, letL"WD fs1; s2; : : : ; s2h."/Ws1< s2< : : : < s2h."/and "si D 0 f or al l i2 f1; 2; : : : ; 2h."/gg.

For alll2L", we have :

lis even H) "iD1for allsl < i < slC1; lis odd H) "iD 1for allsl < i < slC1; "i D1for alli < s1ori > s2h."/.

For example,

˝2D f.1; 1/ ; .0; 0/g;

˝3D f.1; 1; 1/ ; .1; 0; 0/ ; .0; 0; 1/ ; .0; 1; 0/g;

˝4D f.1; 1; 1; 1/; .0; 0; 1; 1/; .1; 0; 0; 1/; .1; 1; 0; 0/; .0; 0; 0; 0/; .0; 1; 0; 1/;

.1; 0; 1; 0/; .0; 1; 1; 0/g:

The lines of the following table represent the elements of˝5,

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1 1 1 1 1 0 1 1 0 1

0 0 1 1 1 1 0 1 1 0

1 0 0 1 1 0 1 1 1 0

1 1 0 0 1 0 0 0 0 1

1 1 1 0 0 0 0 1 0 0

0 1 0 1 1 1 0 0 0 0

1 0 1 0 1 0 1 0 0 0

1 1 0 1 0 0 0 0 1 0

The lines of the following table represent the elements of˝6,

1 1 1 1 1 1 0 0 0 0 1 1

0 0 1 1 1 1 0 0 1 0 0 1

1 0 0 1 1 1 0 0 1 1 0 0

1 1 0 0 1 1 1 0 0 0 0 1

1 1 1 0 0 1 1 0 0 1 0 0

1 1 1 1 0 0 1 1 0 0 0 0

0 1 0 1 1 1 0 1 0 0 0 1

1 0 1 0 1 1 0 1 0 1 0 0

1 1 0 1 0 1 1 0 1 0 0 0

1 1 1 0 1 0 1 0 0 0 1 0

0 1 1 0 1 1 0 0 1 0 1 0

1 0 1 1 0 1 0 0 0 1 0 1

1 1 0 1 1 0 0 1 0 0 1 0

0 1 1 1 0 1 0 0 0 1 1 0

1 0 1 1 1 0 0 1 1 0 0 0

0 1 1 1 1 0 0 0 0 0 0 0

Theorem 2. For any positive integers ri .1i s/ and any integers2, we have

Fr1Cr2CCrsC1D X

."1;"2;:::;"s/2˝s

s

Y

iD1

FriC"i; (3.1)

Proof. LetPr1Cr2CCrsbe a path withr1Cr2C Crs vertices. We subdivide Pr1Cr2CCrs in consecutive blocs of pathsPri withri vertices.1is/ ;see Fig- ure1.

In one hand side, by Theorem1, we have Z.Pr1CCrs/DFr1CCrsC1: In the other handZ.Pr1Cr2CCrs/is the number of independent edge subsets in

Pr1Cr2CCrs:So,Z.Pr1Cr2CCrs/DPs 1

iD0jMkjwhereMk is the set of independ- ent edge subsetsI inPr1Cr2CCrs such that it exists exactlykedges between blocs of pathsPri .1i s/which belong toI.

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v1 v2 vr1 vr1C1 vr1Cr2

Pr1 Pr2

vr1C:::Crs 1C1

Prs

vr1C:::Crs

FIGURE1. PathPr1Cr2C:::Crs divided in consecutive blocs of paths Pri withri vertices.1i s/.

M0 is the set of independent edge subsets I inPr1Cr2CCrs such that doesn’t contain any edge between blocs of pathsPri .1is/, so all these independent edge subsets are in blocsPri .1is/:Hence,jM0j DQs

iD1FriC1:

M1 is the set of independent edge subsetsI inPr1Cr2CCrs such that it exists only one edge between blocs of pathsPri .1is/which belong toI. LetH be a subset ofM1containing the edgevr1CCrkvr1CCrkC1.1ks 1/in all of its independent edge subsets. We contract the adjacent verticesvr1CCrk; vr1CCrkC1

inPr1CCrs into one vertexv0andPr1CCrs 1is a new path after contraction com- posed of consecutive blocs of pathsPr1; Pr2; : : : ; Prk 1; Prk 1; v0; PrkC1 1; PrkC2; : : : ; Prs. A pathPr1CCrs 1does not contain any edge between blocs which belong to independent edge subsets of H, sojHj D Fr1C1Fr2C1 Frk 1C1Frk F2FrkC1FrkC2C1 FrsC1. Thus,jM1j DP

."1;"2;:::;"s/21

Qs

iD1FriC"i

where1is the set of."1; "2; :::; "s/such that for1is; "i 2 f0; 1gand"1"2 "s

forms a sequence such that there is only one pair of zeros and this pair is of the form ."l; "lC1/.

M2 is the set of independent edge subsetsI inPr1Cr2CCrs such that it exists exactly two edges between blocs of pathsPri .1is/which belong toI:As for computing ofjM1jand using the contraction method for the two edges between blocs of pathsPri .1is/;we havejM2j DP

."1;"2;:::;"s/22

Qs

iD1FriC"i where2

is the set of."1; "2; :::; "s/2˝ssuch that for1is; "i 2 f 1; 0; 1gand"1"2 "s

forms a sequence such that there is only one pair of zeros˚

"i; "iC2 and"iC1D 1;

or only two pairs of zeros˚

"i; "iC1

"iC1Ck; "iC2Ck .1kandiCkC2s/and

"l D1for alll 2 f1; 2; :::; i 1; iC2; iC3; :::; iCk; iCkC3; :::; sg.

ForMk .3ks 2/;using the contraction method forkedges between blocs of paths Pri .1 i s/; we have jMkj DP

."1;"2;:::;"s/2k

Qs

iD1FriC"i where k D f."1; "2; :::; "s/2˝sWf or al l si 2L".1i 2h."//;Ph."/

lD1.s2l s2l 1/D kgwhich represents all sequences of˝s such that the sum of the difference of the position of each pair of0is equal tok.

We finish byMs 1which is the set of matchings I inPr1Cr2CCrs such that it exists exactlys 1 edges between blocs of paths Pri .1is/ which belong to

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I. In this case, except the pathsPr1; Prs that lose one vertex after a contraction all others paths Pri .2i s 1/lose two vertices after contraction method. Thus, jMs 1j DFr1Frs

Qs 1 iD2Fri 1:

Note thatfk W1ks 1gis a partition of˝s:Hence, the identity (3.1) holds.

The following corollaries are the main results given by [8].

Corollary 1. For any non-negative integersrandt, we have

FrCt DFrC1FtCFrFt 1: (3.2) Proof. From Theorem2withsD2and˝2D f.1; 1/ ; .0; 0/g, we obtain the fol- lowing identity Fr1Cr2C1DFr1C1Fr2C1CFr1Fr2:We putrDr1 andtDr2C1

and we conclude.

Corollary 2. For any non-negative integersu; vandw, we have FuCvCwDFuC1FvC1FwC1CFuFvFw Fu 1Fv 1Fw 1:

Proof. From Theorem2withsD3and˝3D f.1; 1; 1/; .1; 0; 0/; .0; 0; 1/; .0; 1; 0/g, we obtain the following identityFr1Cr2Cr3C1DFr1C1Fr2C1Fr3C1CFr1C1Fr2Fr3C Fr1Fr2Fr3C1CFr1Fr2 1Fr3: We put uDr1, v Dr2 and wDr3C1 and using Ft DFtC1 Ft 1, we have :

FuCvCwDFuC1FvC1FwCFuC1FvFw 1CFuFvFwCFuFv 1Fw 1

DFuC1FvC1.FwC1 Fw 1/CFuC1FvFw 1CFuFvFw

C.FuC1 Fu 1/ Fv 1Fw 1

DFuC1FvC1FwC1 FuC1FvC1Fw 1CFuC1FvFw 1CFuFvFw

CFuC1Fv 1Fw 1 Fu 1Fv 1Fw 1

DFuC1FvC1FwC1CFuFvFw Fu 1Fv 1Fw 1CFuC1Fv 1Fw 1 CFuC1FvFw 1 FuC1FvC1Fw 1

DFuC1FvC1FwC1CFuFvFw Fu 1Fv 1Fw 1

CFuC1.Fv 1CFv FvC1/ Fw 1

DFuC1FvC1FwC1CFuFvFw Fu 1Fv 1Fw 1:

Corollary 3. For any non-negative integersa; b; candd, we have

FaCbCcCdC1

DFaC1FbC1FcC1FdC1CFaFbFcFdCFaC1FbFcFdC1CFaC1FbC1 CFcFdCFaFb 1FcFdC1CFaFbFcC1FdC1

CFaC1FbFc 1FdCFaFb 1Fc 1Fd:

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Proof. From Theorem2, withsD4 and˝4D f.1; 1; 1; 1/; .0; 0; 1; 1/; .1; 0; 0; 1/;

.1; 1; 0; 0/; .0; 0; 0; 0/ ; .0; 1; 0; 1/; .1; 0; 1; 0/; .0; 1; 1; 0/g, we obtain the

identity.

The following theorem is another identity of Fibonacci number which gives an equivalent of Theorem2.

Theorem 3. Lets2be an integer. For any non-negative integerri .1is/ ; we have

FPs

iD1riC1DFPs 1

iD1riC1FrsC1C

s 2

X

iD0

2 4 0

@

i

Y

jD1

Frs j 1

1

AFPs i 2

jD1 rjC1Frs i 1Frs

3 5:

Proof. As mentioned in Theorem2,FPs

iD1riC1DP

."1;"2;:::;"s/2˝s

Qs

iD1FriC"i: ThenFPs

iD1riC1Dc1Cc0 wherec1 corresponds to the case"s D1andc0 to the case"sD0. That means to countFPs

iD1riC1we have two cases.

Case 1."sD1:Then for all s-uplet."1; "2; :::; "s/we obtain

c1DFrsC1

0

@

X

."1;"2;:::;"s 1/2˝s 1

s 1

Y

iD1

FriC"i

1 A;

so for"sD1we havec1DFPs 1

iD1riC1FrsC1:

Case 2. "sD0:Let"s i 1D0withi the smallest integerk; 0ks 2;such that"s k 1D0. So for1j i we have"s j D 1:Hence,

c0D

s 2

X

iD0

2 4

0

@

i

Y

jD1

Frs j 1

1

AFrs i 1Frs

X

."1;"2;:::;"s i 2/2˝s i 2

s i 2

Y

jD1

FrjC"j

3 5

D

s 2

X

iD0

2 4

0

@

i

Y

jD1

Frs j 1

1

AFPs i 2

jD1 rjC1Frs i 1Frs

3 5:

As an immediate consequence of Theorem3we have :

Corollary 4. For any non-negative integerssandr, we have

Fs rC1DFrC1F.s 1/rC1CFr2

s 2

X

iD0

Fr 1i F.s i 2/rC1:

Proof. Use Theorem3withr1Dr2D DrsDr:

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ACKNOWLEDGEMENT

The authors wish to express their grateful thanks to the anonymous referee for her/his comments and suggestions towards revising this paper.

REFERENCES

[1] H. Belbachir and F. Bencherif, “Linear recurrent sequences and powers of a square matrix,”In- tegers, vol. 6, pp. A12, 17, 2006.

[2] J. A. Bondy and U. S. R. Murty, “Graph theory with applications,”North-Holland, 1976.

[3] O. Chan, I. Gutman, T. K. Lam, and R. Merris, “Algebraic connections between topological in- dices,”J. Chem. Inform. Comput. Sci., vol. 38, pp. 62–65, 1998.

[4] H. Deng, “The largest Hosoya index of.n; nC1/-graphs,”Comput. Math. Appl., vol. 56, no. 10, pp. 2499–2506, 2008, doi:10.1016/j.camwa.2008.05.020.

[5] I. Gutman, “Acyclic systems with extremal H¨uckel -electron energy,” Theoret. Chim. Acta, vol. 45, no. 1, pp. 79–87, 1977, doi:10.1007/BF00552542.

[6] I. Gutman and S. J. Cyvin, “Hosoya index of fused molecules,”MATCH Commun. Math. Comput.

Chem., no. 23, pp. 89–94, 1988.

[7] I. Gutman and O. E. Polansky,Mathematical concepts in organic chemistry. Springer-Verlag, Berlin, 1986. doi:10.1007/978-3-642-70982-1.

[8] C. J. Hillar and T. Windfeldt, “Fibonacci identities and graph colorings,”Fibonacci Quart., vol.

46/47, no. 3, pp. 220–224, 2008/09.

[9] H. Hosoya, “Topological index, a newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons,”Bull. Chem. Soc. Jap., vol. 44, no. 9, pp. 2332–

2339, 1971.

[10] T. Koshy,Fibonacci and Lucas numbers with applications, ser. Pure and Applied Mathematics (New York). Wiley-Interscience, New York, 2001. doi:10.1002/9781118033067.

Authors’ addresses

Hac`ene Belbachir

USTHB, Faculty of Mathematics, RECITS Laboratory, DG-RSDT, Po. Box 32, El Alia, 16111, Algiers, Algeria

E-mail address:hbelbachir@usthb.dz or hacenebelbachir@gmail.com

Hakim Harik

USTHB, Faculty of Mathematics, RECITS Laboratory, DG-RSDT, Po. Box 32, El Alia, 16111, Algiers, Algeria

Current address: CERIST, 5 Rue des fr`eres Aissou, Ben Aknoun, Algiers, Algeria E-mail address:hhakim@mail.cerist.dz, harik hakim@yahoo.fr

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