The Electronic Journal of Linear Algebra.
A publication of the International Linear Algebra Society.
Volume 6, pp. 62-71, March 2000.
ISSN 1081-3810. http://math.technion.ac.il/iic/ela
ELA
TIGHT BOUNDSON THEALGEBRAIC CONNECTIVITY OFA BALANCEDBINARY TREE
JASON J. MOLITIERNO
y
, MICHAEL NEUMANN y
,AND BRYAN L. SHADER z Abstrat. Inthispaper,quitetightlowerandupperboundsareobtainedonthealgebrai on-netivity,namely,theseond-smallesteigenvalueoftheLaplaianmatrix,ofanunweightedbalaned binarytreewithklevels andhenen=2
k
1 verties. Thisisaomplishedbyonsideringthe inverseofamatrixoforderk 1readilyobtainedfromtheLaplaianmatrix. Itisshownthatthe algebraionnetivityis1=(2
k
2k+3)+O(1=2 2k
).
Keywords.Binarytrees,Laplaianmatrix,Algebraionnetivity AMSsubjet lassiations.5C50,15A48
1. Introdution. Inthispaperweimproveknownlowerandupperbounds on the algebrai onnetivity, namely, the seond-smallest eigenvalue of the Laplaian matrix
1
, of the balaned binary tree B k
of k levels and hene n =2 k
1 verties. Speially, in [9, Lemma 6.1℄,Guatteryand Millerquote anearlierresult oftheirs in[8℄inwhihtheyhaveshownthatthealgebraionnetivity,(B
k ) ,ofB
k
satises 1
n
(B k
) 2 n : Ournewupperboundis
( B k
)
1 ( 2
k
2k+3)
2k 2 2
k 1 1 andournewlowerbound is
1 (2
k
2k+2) 2k
p
2 2k 1 2 k 1
2
k 1
p 2( 2
k 1 1)
+
1 3 2
p 2os
2k 1
( B k
):
We omment that for largek the dierene betweenthe denominators in the lower boundandtheupperbound isapproximately
1 p
2 2
p 2
+ 1 3 2
p 2
= 1+ p
2 = 2:4142136:::;
Reeived by the editors on 8 September 1999. Aepted for publiation on 16 Marh 2000. Handlingeditor:StephenJ.Kirkland
y
Departmentof Mathematis,UniversityofConnetiut, Storrs,Connetiut06269-3009,USA (jjmolmath.uonn.edu,neumannmath.uonn.edu). Theworkoftheseondauthorwassupported inpartbyNSFGrantNo. DMS9973247.
z
Department of Mathematis, University of Wyoming, Laramie, Wyoming 82071-3036, USA (bshaderuwyo.edu).
1
ELA
AlgebraiConnetivityofaBalanedTree 63 whihimpliesthat the algebraionnetivityofthe balanedbinary treeis 1=(2
k 2k+3)+O(1=2
2k ).
Weproveour main resultsin Setion 3, while in Setion 2we review neessary preliminariesandalsodesribesomemotivationforourworkhere.
We omment that twoother lower estimatesfor (B k
) an be obtainedas spe-ialasesoftwogenerallowerestimatesonthealgebraionnetivityof agraphG. Therstlowerestimate,equaling1=[2(2
k 3)
2
℄, isduetoFriedland[6,Theorem2.6℄ and involvesthe so-alledCheeger lower bound. The seond lowerbound,equaling 3
p
9 1=(2 k 1
1) 2
, is due to Berman and Zhang [1, Theorem 2.2℄. It anbe readilyheked that our newlowerbound is better than thetwobounds just men-tioned.
2. Preliminaries. LetGbeagraphwithverties1,2,...,n. Denotethedegree ofvertexibydeg(i). TheLaplaian(matrix)ofG isthennmatrixL=[`
ij ℄with
` ij
= 8 < :
deg(i) ifi=j,
1 ifi6=j andiisadjaenttoj, 0 otherwise:
The Laplaian L of a graph G is a useful algebrai tool for assessing ertain properties of the graph. Perhapsthe mostwell-known property ofL is the matrix-tree theorem due to Cayley [2℄ (see also Chaiken [3℄), whih relates the Laplaian to the numberof spanning trees of G. Numerous other properties of G, related to the onnetivity and the isoperimetri number of G, are reeted by the spetrum of L (see [14, 16, 17℄ and thereferenes therein). Sine L is asymmetri, positive semidenite, andsingularmatrix, itseigenvaluesare nonnegativereal numbers,and sotheyanbearrangedinnondereasingorder:
0 = 1
2
n
:
Fiedler [5℄ observedthat the seond-smallesteigenvalue, (G) := 2
, of L provides a measure of onnetivity and he alled (G) the algebrai onnetivity of G. In partiularhehasshownthat(G)>0ifandonlyifGisaonnetedgraph.
The algebrai onnetivity of a graph G and its appliations have been exten-sivelystudiedintheliterature;weitethefollowingpapersandthereferenesquoted therein: GroneandMerris[7℄,Merris[14,15℄,Powers[19℄,Pothen,Simon,andLiou [18℄,GuatteryandMiller[9℄,and Kirkland,Neumann,andShader[12,11,13℄.
ELA
64 J.J.Molitierno,M.Neumann,andB.L.Shader propersubsetX ofvertiesofG wedenetheutquotientof X tobe
jXj min fjXj;jXjg
;
wherejXjisthenumberofedgesinGwithexatlyonevertexinX. Theminimum oftheutquotientsistheisoperimetri number,i(G), ofG. Itiswellknownthat
i(G) (G)
2 :
Thebest thresholdutmethod strivesto ndaset X forwhih theutquotientof X isloseto theminimumi(G). This isahievedasfollows.
(i) Assoiatewitheahvertex,i,thevalueoftheithentryofu.
(ii) Sort the verties aordingto their value. For eah index 1 i n 1, omputetheutquotientfortheseparatorobtainedbysplittingtheverties into thosewithsorted indexlessthani andthose withsortedindex greater thani.
(iii) Choosethesplitfrom(ii)that providesthesmallestutquotient.
Asnotedin[9℄,untilreently,therehasnotbeenarigorousanalysisofthequality ofseparatorsproduedbysuhalgorithms. In[9℄,theompletebalanedbinarytrees are used as building bloks to onstrut graphs for whih the best threshold ut methoddoespoorly. Anessentialingredientintheirwork istheirbound
1 n
(B k
) 2 n
onthealgebraionnetivityforaompletebalaned binarytree. 3. TightBounds on (B
k
). Asbefore,letB k
denotethebalanedbinarytree with k 2levelssothat B
k
has n :=2 k
1 verties. We nowrelabel the verties ofB
k
sothat 1isthe root vertex, thevertiesontheleft branh of1preede those ontherightbranh,and thevertiesontheith levelpreedethoseonleveli+1for i=2;3;:::;k 1. Forexample,fork=3,wehavethelabelingillustratedbelow.
q q q
q q q
q 3 4 6 7
2 5
1
Thefollowingnotationisusedthroughoutthissetion. IfN isanmmmatrix, eah of whose eigenvaluesis real, then the eigenvalues of N are denoted by
i (N), i=1;2;:::;m,where
ELA
AlgebraiConnetivityofaBalanedTree 65 ThetraeandadjointofN aredenoted bytr(N)andadj(N),respetively.
Webeginbyshowingthattheproblemofdetermining(B k
)anbetransformed tothatofdeterminingthesmallesteigenvalueofaprinipalsubmatrixoftheLaplaian oforder2
k 1 1.
Proposition3.1. LetL k
be theLaplaian matrixof B k
whereL k
(1;1) isthe prinipal submatrix of L k
obtainedby deleting itsrstrow and olumn.
Proof. NotethatL k
hastheform 2
1. It follows that eah eigenvalue of L k
(1;1) hasmultipliityatleasttwoandhene,bytheCauhyinterlaingproperty(see[10℄), 0 otherwise,
anddenethe(k 1)(k 1)tridiagonalmatrix
F
ConerningF k 1
welaimthefollowingproposition. Proposition3.2. LetL
k =[`
i;j
℄ bethe Laplaian matrixof B k
ELA
66 J.J.Molitierno,M.Neumann,andB.L.Shader (b)eaheigenvalueof F
k 1
isan eigenvalueof L k
(1;1), ()eaheigenvalueof F
k 1 isreal, (d)
1 (F
k 1 ) =
1 (L
k
(1;1)),and (e)tr(F
1 k 1
) = 2 k
(k+1).
Proof. Part(a)followsfromtheobservationthatfor1ijk, (i)Ifjj ij>1,thenthereare noedges joiningavertexin leveli ofB k
to avertex inlevelj.
(ii)If j i=1,theneahvertexin leveli ofB k
isjoined toexatlytwovertiesin levelj.
(iii)Ifj i= 1,theneahvertexinleveliof B k
isjoinedto exatlyonevertexin levelj.
(iv)Ifvisin leveli,then
` v;v
= (
2 ifi=1, 1 ifi=k, 3 otherwise. Now let (;x) bean eigenpair of F
k 1
. Then (a) implies that L k
(1;1)(Qx) = (Qx). Sine the olumns of Q are linearly independent, it follows that Qx 6= 0. Hene (;Qx)is aneigenpairofL
k
(1;1)and so(b)holds. Part() followsfrom (b) and thefatthat L
k
(1;1) isarealsymmetri matrix. Adopting thenotationin the proof of Proposition 3.1, we seethat L
k
(1;1) is thediret sum oftwoopies of the matrixC. Sine Cis aprinipalsubmatrix ofaLaplaianmatrixand isirreduible, CisanonsingularM-matrix. HeneChasanonnegativeeigenvetorvorresponding totheeigenvalue
1
(C)= 1
(L k
(1;1)). Itisnoweasyto verifythat
[v T
0℄L k
(1;1) = 1
(L K
(1;1))[v T
0℄: Thefatthatvisanonnegativeeigenvetorimpliesthat[v
T
0℄Q6=0. Thisalongwith (a)impliesthat[v
T
0℄Qisaleft-eigenvetorofF k 1
orrespondingtotheeigenvalue
1 (L
k
(1;1)). Part(d)nowfollowsfrom(b).
Finally, we prove(e). As an be readilyheked, the matrix F k 1
admits the following fatorization as the produt of an upper triangular matrix and a lower triangularmatrix:
F k 1
= 2 6 6 6 6 6 6 6 6 6 6 6 6 4
1 2 0 :: :: 0 0 1 2 0 0
0 1 2 0 1 :: .
. .
. .
.
:: ::
. . . .
. .
:: 2 0 0 1 2
3 7 7 7 7 7 7 7 7 7 7 7 7 5
2 6 6 6 6 6 6 6 6 6 6 6 4
1 0 :: :: 0 1 1
1 1 :: ::
:: ::
. . . :: ::
:: 1 0 1 1
ELA
AlgebraiConnetivityofaBalanedTree 67 andthuswehavetheLU-fatorization
F ItfollowsfromProposition3.2thattoeitherdetermineexatlyorapproximately
1
(L(1;1)),itsuÆestoonsidertheproblemofdeterminingexatlyorapproximately thesmallesteigenvalueofF
k 1
. Webeginbynotingthat foranym1,ifD isthe mmdiagonalmatrixwhose(i;i)thentryis2
i=2
is(diagonally)similar toF m
andheneithasthesameeigenvaluesasF m
(m;m) be the leadingprinipal submatrix of G m
of order m 1. Note that S
m 1
is a symmetri, tridiagonal matrixeah of whose diagonal entriesis3andeahofwhosesuper-andsubdiagonalentriesis
p
2. Someadditional propertiesofthematriesS
1 Assume that t 2and proeed by indution. Note that by Laplaeexpansion alongthelastrow,
ELA
68 J.J.Molitierno,M.Neumann,andB.L.Shader NextnotethatifwesetdetS Statement(b)nowfollowsfrom(a).
We shall now use Lemma 3.3 to obtain a lower bound on 1
(G m
). Sine the eigenvaluesofS
m 1
interlaetheeigenvaluesofG m
,andastheeigenvaluesofG m FurthermoreG
m
isanonsingularM-matrix;eahofitseigenvaluesispositive,asare thoseofS
m 1
. HenetheeigenvaluesofF 1 m
interlaethoseofS 1 Henewehavethat
tr(F
ELA
AlgebraiConnetivityofaBalanedTree 69
Proof. ReallthatthattheeigenvaluesofG m
arethereiproalsoftheeigenvalues of thematrixF
1 m
usedabove. Thus (4)followsfrom (3) and thetraeformulasin Proposition3.2 (withm=k 1)andLemma 3.3b(witht=m 1).
To obtain an upper bound on 1
(G m
), we onsider the following tt matrix whih, for t = m, is readily seen to be a perturbation by a positive semidenite matrixofG
m
Thus,fromWeyl'stheoremontheperturbationofeigenvaluesofsymmetrimatries (seep.181ofHornandJohnson[10℄),wehavethat
Weomment that from Elliot [4℄, itan be deduedthat theeigenvaluesof H t aregivenby
WeproeednowtoproveforH t
aresultsimilarto Lemma3.3. Lemma 3.5. ForH
t
asgiven in(5), thefollowing onditionshold: (a)det(H thedeterminantweseethatdet (H
t
). Thuspart(a)follows fromtheformulaforthedeterminantofS
t 1 andS
t
,whihisgiveninLemma 3.3a. Fort2,the(i;i)thentryofadj(H
ELA
70 J.J.Molitierno,M.Neumann,andB.L.Shader To obtain an upper bound on
1 (G
m
), we one again resort to working with inverses. From(6) andthe fatthat theeigenvaluesofF
m
arethesameasthose of G
m
,wehavethat
i (F
1 m
) i
(H 1 m
); i=1;2;:::;m: Thisimpliesthat
m
(F 1 m
)+tr(H 1 m
) m
(H 1 m
) tr(F 1 m
):
Substituting intheappropriatetraeformulasfrom Lemmas3.3and3.5yields
m (F
1 m
) 2 m+1
2m
2m+2 p
2( 2m+1 2 m
) 2
m+1 1
p 2(2
m 1)
+ m
(H 1 m
): Ifwenowmakeuseofthefat that
m (H
1 m
)isthereiproalofthesmallest eigen-valueofH
m
, wearrive,using(7),at thefollowinglowerbound onthe 1
(G m
). Lemma 3.6. LetG
m
bethematrix givenin (2)with m3. Then
1 (G
m )
1 2
m+1 2m
2m+2 p
2( 2m+1 2 m
) 2
m+1 1
p 2(2
m 1)
+
1 3 2
p 2os
2m+1
:
Thegoalofourpaperistoprovidegoodlowerandupperbounds on (B
k ) =
1 (L
k
(1;1)) = 1
(F k 1
) = 1
(G k 1
):
Thus,by usingLemmas 3.4and 3.6(withm=k 1), weobtainthemain resultof thispaper.
Theorem 3.7. LetB k
bethe balanedbinarytreeonk4levels. Then (B
k )
1 ( 2
k
2k+3)
2k 2 2
k 1 1 and
(B k
)
1 (2
k
2k+2) 2k
p
2 2k 1 2 k 1
2
k 1
p 2( 2
k 1 1)
+
1 3 2
p 2os
2k 1
ELA
AlgebraiConnetivityofaBalanedTree 71 REFERENCES
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