Simplices and spectra of graphs, continued

0.08 MB·English

by  Bojan Mohar

SIMPLICES AND SPECTRA OF GRAPHS, CONTINUED
BOJANMOHARANDIGORRIVIN
9
0 Abstract. Inthisnoteweshowthatthe(n−2)-dimensionalvolumesofcodimen-
0 sion2facesofann-dimensionalsimplexarealgebraicallyindependentquantities
2 ofthevolumesofitsedge-lengths. Theproofinvolvescomputationoftheeigen-
n valuesofKnesergraphs.
a
J

0
2 ntroduction
I

]
G Let T be the set of congruence classes of n-simplices in Euclidean space En.
n
M The set T is an open manifold (also a semi-algebraic set) of dimension n+1 .
n 2
h. Coincidentally, a simplex T ∈ Tn is determined by the n+21 lengths of its ed(cid:0)ges(cid:1).
t Furthermore, the square ofthevolume ofT ∈ Tn isapol(cid:0)yno(cid:1)mial inthe squaresof
ma theedge-lengthsℓij = kvi−vjk2 (1 ≤ i < j ≤ n+1),wherev1,...vn+1 arethevertices
of T. This polynomial is given by the Cayley-Menger determinant formula (cf.,
[
e.g., [5]or[2]):
1
v (−1)n+1
4 (1) V2(T) = detC,
8 2n(n!)2
2
3 whereCistheCayley-Mengermatrixofdimension(n+2)×(n+2),whoserowsand
. columns are indexedby {0,1,...,n+1}andwhose entriesare definedasfollows:
1
0
9 =
0, i j
0
: C = 1, ifi = 0 or j = 0, andi , j
v ij 

i ℓ2 , otherwise.
X  ij

ar Notethatann-simplexhasn+21 edgesandthesamenumberof(n−2)-dimensional
faces, and sothe following question isnatural:
(cid:0) (cid:1)
Date:January21,2009.
SupportedinpartbytheARRSResearchProgramP1-0297,byanNSERCDiscoveryGrantand
by the Canada Research Chair program. On leave from Department of Mathematics, IMFM &
FMF,UniversityofLjubljana,Ljubljana,Slovenia.
TheauthorwouldliketothanktheAmericanInstitute ofMathematicsforaninvitationtothe
workshopon“RigidityandPolyhedralCombinatorics”,wherethisworkwasstarted. Theauthor
hasprofitedfromdiscussionswithIgorPak,EzraMiller,andBobConnelly.
1
2 BOJANMOHARANDIGORRIVIN
Question 1. Is the congruence class of every n-simplex determined by the (n−2)-
dimensional volumesof its (n−2)-faces?
Question 1 must be classical, but the earliest reference stating it that we are
aware ofis WarrenSmith’s PhDthesis [10].
AttheAIMworkshoponRigidityandPolyhedralCombinatorics,BobConnelly
(who wasunaware ofthe reference [10])raised the following:
Question2. Isthevolumeofeveryn-simplexdeterminedbythe(n−2)-dimensional
volumes ofits (n−2)-faces?
=
In fact, Connelly stated Question 2 for n 4, which is the first case where the
question is open. For n = 3 the answer is trivially ”Yes”, since 3 −2 = 1, and we
are simply asking if the volume of the simplex is determined by its edge-lengths.
In dimension 2, the answer is trivially ”No”, since 2 − 2 = 0, and the volume of
codimension-2 facesofa triangle carriesnoinformation.
ffi ffi
Clearly,thea rmativeanswertoQuestion1wouldimplyana rmativeanswer
toQuestion2. InthispaperwefirstshowthattheanswertoQuestion2,andhence
also to Question 1 is negative for every n ≥ 4. We actually found out that this
=
has been answered previously for n 4 in [1], where an example is given and
attributed to Philip Tuckey; seealso [3].
Our examples are given in a separate section. Several reasons suggest that the
ffi
following question maystill havean a rmative answer:
Question 3. Is it true that for every choice of n+1 positive real numbers, there are
2
only finitely many congruence classes of n-simplices whose (n − 2)-dimensional
(cid:0) (cid:1)
volumes ofthe (n−2)-facesare equalto these numbers?
Inthis note we show thata weakerstatement holds:
Theorem 4. The n+1 (n− 2)-dimensional volumes of the (n−2)-faces of an n-simplex
2
are algebraicallyindependentoverC[ℓ ; 1 ≤ i < j ≤ n+1].
(cid:0) (cid:1) ij
Theorem4isclearlyanecessarystepinthedirectionofresolvingQuestion3,but
is far from sufficient. To show it, consider the map of R(n+1)n/2 to R(n+1)n/2, which
sends the vector ℓ of edge-lengths of an n-simplex to the vector Y of volumes
of (n − 2)-dimensional faces. To show Theorem 4, it is enough to check that the
=
Jacobian J(ℓ) ∂Y/∂ℓ is non-singular at one point. We will use the most obvious
point p , the one corresponding to a regular simplex with all edge-lengths equal
1
=
to 1. By symmetry considerations, the Jacobian J(p ) can be written as J(p ) cM,
1 1
where cis aconstant andMis
1, ifthe edgee isincidentwith the (n−2)-face F
=
M
e,F
(0, otherwise.
SIMPLICESANDSPECTRAOFGRAPHS, CONTINUED 3
The first observation isthatthe constant cabove isnot equalto 0:
1
=
Lemma5. J(p ) M.
1 (n−2)!(n−1)1/22(n−4)/2
(n−1)1/2
Proof. Letν = denotethe(n−2)-dimensionalvolumeoftheregular
(n−2)!2(n−2)/2
(n − 2)-simplex with all edge-lengths 1. Let us observe that the volume of a k-
dimensionalsimplexisahomogeneousfunctionofdegreekoftheedge-lengths. An
application of Euler’sHomogeneous Function Theorem shows thatat p ,
1
∂Y 2 ν, ifthe edge eisincident with the (n−2)-face F
F = n−1
∂ℓ (0, otherwise.
e
Thisimpliesthat c = 2 ν andcompletes the proof. (cid:3)
n−1
heeigenvalues of
T M
Asshownabove,Theorem4reducestotheassertionthatthedeterminantofthe
matrix M is not zero. We will actually be able to compute all eigenvalues of M,
which isof interest in itsown right.
Theorem6. Eigenvaluesof Mareλ = n−1 (simpleeigenvalue),λ = 1withmultiplic-
1 2 2
ity 12(n+1)(n−2), and λ3 = 2−nwith(cid:0)mul(cid:1)tiplicityn.
Corollary 7. Theabsolute valueof the determinantof Mequals
1(n−2)n+1(n−1) , 0,
2
for n > 2.
To prove Theorem 6, let us first observe that the n+1 rows of M are indexed by
2
the2-elementsubsetsofthesetR = {1,...,n+1},anditscolumnsareindexedbythe
(cid:0) (cid:1)
(n−1)-subsetsFofR. ByreplacingeachcolumnindexFwithitscomplementR\F,
then the columns are indexed by the same set as the rows. After this convention,
=
thematrixMbecomesasymmetricmatrixwithzerodiagonalsinceM 1ifand
e,f
only if e ⊆ R\ f, which is equivalent to f ⊆ R \ e. Therefore, M is the adjacency
matrix of a graph G whose vertices are the 2-element subsets of R, and two of
n
themareadjacentifandonlyiftheyaredisjoint. Thus,thecomplementG ofG is
n n
+
isomorphic to the line graph L(Kn+1)ofthe complete graph Kn+1 on n 1 vertices.
TheeigenvaluesofL(Kn+1)are(see[4,p.19]): t1 = 2n−2,t2 = −2,andt3 = n−3,
withthesamemultiplicities(respectively)asclaimedabovefortheeigenvaluesof
M. SincethegraphL(Kn+1)isregular,itisaneasyexercisetoseethatitsadjacency
matrix A and the adjacency matrix M of its complement have the same set of
4 BOJANMOHARANDIGORRIVIN
eigenvectors. By using the fact that A + M + I = xt · x, where x = (1,...,1)t is
the eigenvector of A and M corresponding to the dominant eigenvalues of these
matrices,weconcludethattheeigenvaluesofMareλ = n+1 −t −1andλ = −t −1
1 2 1 i i
fori = 2,3(preserving multiplicities). Thus, λ1 = n+21 −(cid:0)2n+(cid:1) 1 = n−21 ,λ2 = 1,and
λ = 2−n, respectively.
3 (cid:0) (cid:1) (cid:0) (cid:1)
ingular examples
S
Let us consider the n-simplex in Rn with vertices v ,v ,...,v given as follows.
0 1 n
The vertex v has the first n − 2 coordinates equal to ((n − 1)1/2 + 1)/(21/2(n − 2)),
0
while its last two coordinates are 0. For i = 1,2,...,n − 2, the vertex v has ith
i
coordinate equal to 2−1/2 and all other coordinates 0. These n − 1 vertices form
a regular (n − 2)-simplex contained in Rn−2 ⊂ Rn with all side lengths 1. Let
a = 1 n−2v be its barycenter, and let c := kv − ak denote the distance from a
n−1 i=0 i 0 2
to the vPertices vi. A short calculation shows that c2 = 12 − 2(n1−1). Now, let vn−1 be
obtained from a by changing its last two coordinates to be real numbers p and q
satisfyingp2+q2 = 1−c2. Similarly,letv beobtainedinthesamewaybychoosing
n
anotherpairr,sofnumberssatisfyingr2+s2 = 1−c2. Thisgivesrisetoann-simplex
whose all sides are equal to 1 except for the side v v whose square length is
n−1 n
t := (p−r)2+(q−s)2. Byfixingt,thissimplexisdetermineduptocongruence,and
we denote it by T(t). Observe that t may take any value between 0 and 4(1−c2),
byselecting p,q,r,sappropriately.
Nextweobservethatthevolumesofthe(n−2)-facesofT(t)takeonlytwovalues.
If an (n − 2)-face does not contain both v and v , then it is a regular simplex,
n−1 n
whosevolumeisindependentoft. Ontheotherhandifan(n−2)-simplexcontains
=
v and v ,its volume w w(t)isuniquelydeterminedby t. In fact, ifwe putthe
n−1 n
square distances in the Cayley-Menger determinant, we conclude that w(t)2 is a
quadratic polynomial in t, w(t)2 = αt2 +βt+γ. If t = 0, the volume is 0, so γ = 0.
For t = 1 we have the regular (n−2)-simplex, so α+β = n−1 . Finally, using
2n−2((n−2)!)2
(1)(withthevalueofnbeingreplacedbyn−2)andlookingattheCayley-Menger
determinantexpansion term with t2, we conclude that
(−1)n−1
α = − det(J −I ),
2n−2((n−2)!)2 n−2 n−2
where J is the all-1-matrix and I is the identity matrix of order n− 2. Since
n−2 n−2
det(J −I ) = (−1)n−3(n−3), we conclude that α = −(n−3)22−n/((n −2)!)2 and
n−2 n−2
β = (n−2)21−n/((n−2)!)2. Inparticular,
1
w(t)2 = ((3−n)t2 +(2n−4)t).
2n−2((n−2)!)2
SIMPLICESANDSPECTRAOFGRAPHS, CONTINUED 5
This function is symmetric around the point t = n−2. Consequently, the non-
0 n−3
congruent n-simplices T(t − x) and T(t + x) have the same (n − 2)-volumes of
0 0
their (n−2)-faces for each admissible value of x, i.e. for 0 < x < n−2−4/(n−1). These
n−3
examplesthusshow that Questions 1 and2 have negative answers.
oncluding remarks
C
One can ask the same question as above for other dimension-complementary
volumes, i.e. about the volumes of the (k − 1)-faces and the (n − k)-faces of an
n-simplex,where2 ≤ k ≤ n/2. Ifonewouldcompare,similarlyasinthecasek = 2
above,thedependenceof(n−k)-volumesofan(n−k)-faceQonthe(k−1)-volumes
ofthe(k−1)-facesF ⊂ Q,thecorresponding“Jacobian”wouldagainbeaconstant
multiple of a symmetric matrix M, whose entries are indexed by the k-subsets
of the set R = {1,...,n + 1} (after the column indices pass to the complementary
subsets), and
1, ifthe E∩F = ∅
=
(2) M
E,F
(0, otherwise.
+
The graph whose adjacency matrix is M is known as the Kneser graph K(n 1,k).
Its eigenvalues can be comuted using the methods from the theory of association
schemesand can befound, forexample, in [8,Section 9.4].
Theorem8. Letnandkbeintegers,where2 ≤ k ≤ n/2,and letMbethematrixoforder
n+1 × n+1 whoseentriesare determinedby(2). The eigenvaluesof Mare theintegers
k k
(cid:0) (cid:1) (cid:0) (cid:1) n−k−i+1
λ = (−1)i , i = 0,1,...,k.
i k−i !
Since 2 ≤ k ≤ n/2, none of the eigenvalues in Theorem 8 is zero. This raises the
question whether there is an analogy with Theorem 4 for 2 ≤ k ≤ n/2, between
the collection of the n+1 (n − k)-dimensional volumes of the (n − k)-faces of an
k
n-simplexand thecollection ofall (k−1)-dimensional volumesofits(k−1)-faces.
(cid:0) (cid:1)
Asa final remark, we would like to point out that our original approach to this
problem [9] used results about divisors [6] (also known as equitable partitions [8])
combined with the representation theory of the symmetric group and the notion
ofGelfand pairsasdevelopedin [7].
eferences
R
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6 BOJANMOHARANDIGORRIVIN
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DepartmentofMathematics,SimonFraserUniversity,Burnaby,B.C.V5A1S6
E-mailaddress: [email protected]
DepartmentofMathematics,TempleUniversity,Philadelphia
E-mailaddress: [email protected]