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