Simplices and spectra of graphs, continued

SIMPLICES AND SPECTRA OF GRAPHS, CONTINUED BOJAN MOHAR AND IGOR RIVIN A bstract . In this note we show that the ( n − 2 )-dimensional vo lumes of c odimen- sion 2 face s of an n -dimensional simplex are algebra ically independent quantities of the volumes of its e dge-lengths. The proof involves computation of the eigen- values of Kneser gra phs. I ntroduction Let T n be the set of congruence classes of n -simplices in Euclidean space E n . The set T n is an open ma nifold (also a semi-algebraic set) of dimension  n + 1 2  . Coincidentally , a simplex T ∈ T n is determined by the  n + 1 2  lengths of its edges. Furthermor e, the square of the volume of T ∈ T n is a polynomial in the squares of the e dge-lengths ℓ i j = k v i − v j k 2 (1 ≤ i < j ≤ n + 1), wher e v 1 , . . . v n + 1 ar e the vertices of T . This polynomial is given by the Cayley-Menger determinant formula (cf., e.g., [5] or [2]): (1) V 2 ( T ) = ( − 1) n + 1 2 n ( n !) 2 det C , wher e C is the Cayley-Menger matrix of dimension ( n + 2) × ( n + 2), whose rows a nd columns ar e indexed by { 0 , 1 , . . . , n + 1 } and whose entries are defined as follows: C i j =            0 , i = j 1 , if i = 0 or j = 0, and i , j ℓ 2 i j , otherwise. Note that an n -simplex has  n + 1 2  edges and the same number of ( n − 2)-dimensional faces, an d so the following question is natural: Date : June 1, 20 18. Supported in pa rt by the A RRS Research Program P1-02 97, by an NSERC Discovery Grant and by the Canada Research Chair program. On leave from Department of Mathematics, IM FM & FMF , University of L jubljana, Ljubljana, Slovenia. The author would like to thank the Amer ica n Institute of Mathematics for an invitation to the workshop on “Rigidity and Polyhedral Combinatorics”, where this work was started. The a uthor has profited from discussions with Igor Pak, Ezra Miller , and Bob Connelly . 1 2 BOJAN MOHAR AND IGOR RIVIN Question 1 . Is the cong ruence class of every n -simplex determined by the ( n − 2)- dimensional volumes of its ( n − 2)-faces? Question 1 must be classical, but the earliest refer ence stating it that we ar e aware of is W arr en Smith’s PhD thesis [10]. At the AIM workshop on Rigidity and Polyhedral Combinatorics, B ob C onnelly (who was unaware of the refer ence [10]) raised the following: Question 2 . Is the v olume of every n -simplex determined by the ( n − 2)-dimensional volumes of its ( 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 ”Y es”, since 3 − 2 = 1 , and we ar e 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 faces of a triangle carries no informatio n. Clearly , the a ffi rmative answer to Question 1 would imply an a ffi rmative a nswer to Question 2. In this pape r we first show that the answer to Question 2, and he nce also to Question 1 is negative for every n ≥ 4. W e actually found out that this has been answered previously for n = 4 in [1], where an example is given and attributed to Philip T uckey; see also [3]. Our e xample s are given in a separate section. Several reaso ns suggest that the following question may still have an a ffi rmative answer: Question 3 . Is it true that for e very choice of  n + 1 2  positive real numbers, there ar e only finitely many congruence classes of n -simplices whose ( n − 2)-dimensional volumes of the ( n − 2 )-faces ar e equal to these numbers? In this note we show that a weaker statement holds: Theorem 4. The  n + 1 2  ( n − 2) -di mensional volume s of the ( n − 2) -faces of an n-sim plex are algebraically independent over C [ ℓ i j ; 1 ≤ i < j ≤ n + 1] . Theor em 4 is clearly a necessary step in the dir e ction of r esolving Question 3 , but is far fr om su ffi cient. T o 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. T o show Theor e m 4, it is enough to check that the Jacobian J ( ℓ ) = ∂ Y /∂ℓ is non-singular at one point. W e will use the most obvio us point p 1 , the one corr esponding to a r egular simplex with all edge-lengths equal to 1. By symmetry consideratio ns, the Jacobian J ( p 1 ) can be written as J ( p 1 ) = cM , wher e c is a constant and M is M e , F = ( 1 , if the edge e is incident with the ( n − 2)-face F 0 , otherwise. SIMPLICES AND SPECT RA OF GRAPHS, CONTINUED 3 The first observation is that the constant c above is not equal to 0: Lemma 5. J ( p 1 ) = 1 ( n − 2)! ( n − 1 ) 1 / 2 2 ( n − 4) / 2 M. Proo f . Let ν = ( n − 1) 1 / 2 ( n − 2)! 2 ( n − 2) / 2 denote the ( n − 2)-dimensional volume of the r egular ( n − 2)-simplex with all ed ge-lengths 1. L e t us observe that the volume of a k - dimensional simplex is a homoge neous function of degr ee k of the e d ge-lengths. An application of Euler ’s Homogeneous Function Theorem shows that at p 1 , ∂ Y F ∂ℓ e = ( 2 n − 1 ν , if the edge e is incident with the ( n − 2 )-face F 0 , otherwise . This implie s that c = 2 n − 1 ν and completes the proof .  T he eigenv alues of M As shown above, Theorem 4 r ed uces to the assertion that the determinant of the matrix M is not zero . W e will actually be able to compute all ei genvalues of M , which is of inter est in its own right . Theorem 6. Eigenvalues of M ar e λ 1 =  n − 1 2  (simple eigenvalue), λ 2 = 1 with m ultiplic- ity 1 2 ( n + 1)( n − 2) , and λ 3 = 2 − n with multipli city n. Corollary 7. The absolute value of the deter minant of M equals 1 2 ( n − 2) n + 1 ( n − 1) , 0 , for n > 2 . T o prove Theorem 6, let us first observe that the  n + 1 2  r ows of M are indexed by the 2-eleme nt subsets of the set R = { 1 , . . . , n + 1 } , and its columns are indexed by the ( n − 1)-subsets F of R . By replacing e ach column index F with its complement R \ F , then the columns are indexed by the same set as the rows. After this convention, the matrix M becomes a symmetric matrix with zero diagonal since M e , f = 1 if and only if e ⊆ R \ f , which is equivalent to f ⊆ R \ e . Therefor e, M is the adjacency matrix of a graph G n whose vertices are the 2-element subsets of R , and two of them ar e adjacent if and only if they are disjoint. Thus, the complement G n of G n is isomorphic to the lin e graph L ( K n + 1 ) of the complete graph K n + 1 on n + 1 vertices. The eigenvalues of L ( K n + 1 ) a r e (see [4, p. 1 9]): t 1 = 2 n − 2, t 2 = − 2, and t 3 = n − 3, with the same multiplicities (respectively) as claimed above for the e igenvalues of M . Since the graph L ( K n + 1 ) is regular , it is a n e asy exercis e to see that its adjacency matrix A and the adjacency matrix M of its complement have the same set of 4 BOJAN MOHAR AND IGOR RIVIN eigenvectors. By using the fact that A + M + I = x t · x , whe re x = (1 , . . . , 1) t is the eigenvector of A and M corr esponding to the dominant eigenvalues of these matrices, we conclude that the eigenvalues of M are λ 1 =  n + 1 2  − t 1 − 1 and λ i = − t i − 1 for i = 2 , 3 (pr eserving multiplicities). Thus, λ 1 =  n + 1 2  − 2 n + 1 =  n − 1 2  , λ 2 = 1, and λ 3 = 2 − n , r espe ctively . S ingular examples Let us consider the n -simplex in R n with vertices v 0 , v 1 , . . . , v n given as follows. The vertex v 0 has the first n − 2 coor dinates e qual to (( n − 1) 1 / 2 + 1) / (2 1 / 2 ( n − 2)), while its last two coordinates are 0. For i = 1 , 2 , . . . , n − 2, the vertex v i has i th coor dinate e qual to 2 − 1 / 2 and all other coordinat e s 0. These n − 1 vertices form a r egular ( n − 2)-simplex contained in R n − 2 ⊂ R n with all side lengths 1. Let a = 1 n − 1 P n − 2 i = 0 v i be its barycenter , and let c : = k v 0 − a k 2 denote the distance from a to the vertices v i . A short calculation shows that c 2 = 1 2 − 1 2( n − 1) . Now , let v n − 1 be obtained fr om a by changing its last two coordinat e s to be r eal numbers p and q satisfying p 2 + q 2 = 1 − c 2 . Similarly , let v n be obtained in the same way by choosing another pair r , s of numbers satisfying r 2 + s 2 = 1 − c 2 . This gives rise to an n -simplex whose a l l sides ar e equal to 1 except for the side v n − 1 v n whose square length is t : = ( p − r ) 2 + ( q − s ) 2 . By fixing t , this simplex is determined up to congruence, and we de note it by T ( t ). Observe that t may take any value between 0 and 4(1 − c 2 ), by selecting p , q , r , s appr opriately . Next we observe that the volumes of the ( n − 2)-faces of T ( t ) take only two values. If an ( n − 2)-face does not contain both v n − 1 and v n , then it is a r egular simplex, whose volume is inde pendent of t . On the other ha nd if an ( n − 2 )-simplex contains v n − 1 and v n , its volume w = w ( t ) is uniquely d etermined by t . In fact, if we put the squar e distances in the Cayley-Menger determinant, we conclude that w ( t ) 2 is a quadratic polynomial in t , w ( t ) 2 = α t 2 + β t + 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 2 n − 2 (( n − 2)!) 2 . Finally , using (1) (with the value of n being replaced by n − 2) and looking at the Cayley-Menger determinant expansion term with t 2 , we conclude that α = − ( − 1) n − 1 2 n − 2 (( n − 2 )!) 2 det( J n − 2 − I n − 2 ) , wher e J n − 2 is the all-1-matrix and I n − 2 is the identity matrix of or der n − 2. Since det( J n − 2 − I n − 2 ) = ( − 1) n − 3 ( n − 3), we conc lude that α = − ( n − 3)2 2 − n / (( n − 2)!) 2 and β = ( n − 2)2 1 − n / (( n − 2)!) 2 . In particular , w ( t ) 2 = 1 2 n − 2 (( n − 2)!) 2 ((3 − n ) t 2 + (2 n − 4 ) t ) . SIMPLICES AND SPECT RA OF GRAPHS, CONTINUED 5 This function is symmetric ar ound the point t 0 = n − 2 n − 3 . Consequently , the non- congr uent n -simplices T ( t 0 − x ) and T ( t 0 + x ) have the same ( n − 2)-volu me s of their ( n − 2)-faces for e a ch admissible value of x , i.e. for 0 < x < n − 2 − 4 / ( n − 1) n − 3 . These examples thus show that Questions 1 and 2 have negative a nswers. C oncluding remarks One can ask the same question as above for other dimension-complementary volumes, i.e. about the volumes of the ( k − 1)-faces a nd the ( n − k ) -faces of an n -simplex, where 2 ≤ k ≤ n / 2. If one would compare, similarly as in the case k = 2 above, the dependence of ( n − k )-volumes of an ( n − k )-face Q on the ( k − 1)-volumes of the ( k − 1)-faces F ⊂ Q , the corr esponding “ Jacobian” would again be a constant 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 (2) M E , F = ( 1 , if the E ∩ F = ∅ 0 , otherwise. The graph who se adjacency matrix is M is known as the Kneser graph K ( n + 1 , k ). Its eigenvalues can be comuted using the methods fr om the theory of associat ion schemes and can be found, for example, in [8, Section 9.4]. Theorem 8. Let n a nd k be integers, wh ere 2 ≤ k ≤ n / 2 , and le t M be the matrix of order  n + 1 k  ×  n + 1 k  whose entries ar e de termined by (2). The eigenvalues of M are the integers λ i = ( − 1 ) i n − k − i + 1 k − i ! , i = 0 , 1 , . . . , k . Since 2 ≤ k ≤ n / 2, none of the eigenvalues in Theor e m 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 k  ( n − k )-dimensional volumes of the ( n − k )-faces of an n -simplex and the collection of all ( k − 1)-dimensional volumes of its ( k − 1)-faces. As a final remark, we would like to point out that our original appr oach to this pr oblem [9] used results about divis ors [6] (also known as equitable partiti ons [8]) combined wit h the repr esentation theory of the symmetr ic group a nd the notion of Gelf and pairs as developed in [7]. R eferences [1] John W . B a rrett. First order Regge calculus . Class. Quantum Grav . 11 (1 994) 2723 – 2730. [2] Marcel Be rger . Geometry I . Spr inger , Berlin, 1987 . [3] Eugenio Bianchi and Leonardo M ode sto. The perturbative Regge-calc ulus regime of loop quantum gravity . Nuclear Physics B 7 96 [FS] (200 8 ) 581–6 21. 6 BOJAN MOHAR AND IGOR RIVIN [4] Norman Biggs. Algebraic graph th eory , second edition. Cambridge University Press, 200 1. [5] Leonard M. Blumenthal. Theory and applications of distance geometry . Clarendon Press, Oxford, 1953. [6] Drago ˇ s M. Cvetkovi ´ c , M ichael Doo b, and Horst Sachs. Spectra of graph s . Johann A mbrosius Barth, Heidelberg, third edition, 19 95. [7] Persi Diaconis. Group representations in probab ility and stat istics . Institute of Mathematical Statis- tics Lecture Notes—Monograph S e ries, 11 . Institute of Mathematical Statistics, Ha yward, CA, 1988. [8] Chris Godsil and Gordon Royl e . Algebraic graph theory . Gradua te T exts in Mathematics, 207. Springer-V erlag, New Y ork, 2 001. [9] Igor Rivin, Simplices and spectra of grap hs. arXiv:08 03.13 17v1. [10] W arren Douglas Smith. Studies in co mputational geometry m otivated by m esh generation . PhD thesis, Princeton University , 1989 . D ep artment of M a thema tics , S imon F r aser U niversity , B urnaby , B.C. V5A 1S6 E-mail address : moh ar@sfu .ca D ep artment of M a thema tics , T emple U nivers ity , P hiladelphia E-mail address : riv in@tem ple.ed u