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BETTI NUMBERS OF DETERMINANTAL IDEALS ROSA M. MIRO´-ROIG∗ 7 0 Abstract. Let R = k[x1,··· ,xn] be a polynomial ring and let I ⊂ R be a graded 0 ideal. In [16], Ro¨mer asked whether under the Cohen-Macaulay assumption the i-th 2 Betti number βi(R/I) can be bounded above by a function of the maximal shifts in the n minimal graded free R-resolution of R/I as well as bounded below by a function of the a minimal shifts. The goal of this paper is to establish such bounds for graded Cohen- J Macaulayalgebrask[x1,··· ,xn]/I whenI isastandarddeterminantalidealofarbitrary 6 codimension. We also discuss other examples as well as when these bounds are sharp. 1 ] C A . Contents h t a 1. Introduction 1 m 2. Standard Determinantal ideals 3 [ 3. Ideals defined by submaximal minors 8 1 References 14 v 5 3 4 1 1. Introduction 0 7 Let R = k[x ,··· ,x ] be a polynomial ring in n variables over a field k, let deg(x ) = 1 0 1 n i / and let I ⊂ R be a graded ideal of arbitrary codimension. Consider the minimal graded h free R-resolution of R/I: t a m v: 0 −→ ⊕j∈ZR(−j)βp,j(R/I) −→ ··· −→ ⊕j∈ZR(−j)β1,j(R/I) −→ R −→ R/I −→ 0 i X where we denote β (R/I) = dimTorR(R/I,k) the (i,j)-th graded Betti number of R/I i,j i j r and β (R/I) = β (R/I) is the i-th total Betti number. Many important numerical a i j∈Z i,j invariants of I Pand the associated scheme can be read off from the minimal graded free R-resolution of R/I. For instance, the Hilbert polynomial, and hence the multiplicity e(R/I) of I, can be written down in terms of the shifts j such that β (R/I) 6= 0 for some i,j i, 1 ≤ i ≤ p. Let c denote the codimension of R/I. Then c ≤ p and equality holds if and only if R/I is Cohen-Macaulay. We define m (I) = min{j ∈ Z | β (R/I) 6= 0} i i,j Date: February 2, 2008. 1991 Mathematics Subject Classification. Primary 13H15 13D02; Secondary 14M12. ∗ Partially supported by MTM2004-00666. 1 2 ROSAM. MIRO´-ROIG to be the minimum degree shift at the i-th step and M (I) = max{j ∈ Z | β (R/I) 6= 0} i i,j to be the maximum degree shift at the i-th step. We will simply write m and M when i i there is no confusion. If R/I is Cohen-Macaulay and has a pure resolution, i.e. m = M i i for all i, 1 ≤ i ≤ c, then Herzog and Ku¨hl [9] and Huneke and Miller [10] showed that c m e(R/I) = i=1 i Q c! and d β (R/I) = (−1)i+1 j for i = 1,··· ,c. i d −d Yj6=i j i Sincethentherehasbeenaconsiderably efforttoboundthemultiplicity ofahomogeneous Cohen-Macaulay ideal I ⊂ R in terms of the shifts in its graded minimal R-free resolution; and Herzog, Huneke and Srinivasan have made the following conjecture (minimal conjecture): If I ⊂ R be a graded Cohen-Macaulay ideal of codimension c, then c m c M i=1 i ≤ e(R/I) ≤ i=1 i. Q c! Q c! There is a growing body of the literature proving special cases of the above conjecture. For example, it holds for complete intersections [8], powers of complete intersection ideals [7], perfect ideals with a pure resolution [10], perfect ideals with a quasi-pure resolution (i.e. m ≥ M ) [8], perfect ideals of codimension 2 [8], Gorenstein ideals of codimension i i−1 3 [13] and standard determinantal ideals of arbitrary codimension [14]. Another natural question which naturally arises in this context is whether under the Cohen-Macaulay assumption the i-th Betti number β (R/I) can be bounded above by a i function of the maximal shifts in the minimal graded free R-resolution of R/I as well as bounded below by a function of the minimal shifts. In [16], Ro¨mer made a natural guess m m M M j j j j (1.1) ≤ β (R/I) ≤ i m −m m −m M −M M −M 1≤Yj<i i j i<Yj≤c j i 1≤Yj<i i j i<Yj≤c j i for i = 1,··· ,c and he showed that these bounds hold if I is a complete intersection or componentwise linear. Moreover, in these cases we have equality above or below if and only if R/I has a pure resolution. Unfortunately, these bounds are not always valid (see [16]; Example 3.1). For Cohen-Macaulay algebras with strictly quasi-pure resolution (i.e., m > M for all i) he showed i i−1 m m M M j j j j (1.2) ≤ β (R/I) ≤ i M −m M −m m −M m −M 1≤Yj<i i j i<Yj≤c j i 1≤Yj<i i j i<Yj≤c j i for i = 1,··· ,c and again we have equalities if and only if R/I has a pure resolution. Notice that Mj Mj may be negative and thus, in general, 1≤j<i mi−Mj i<j≤c mj−Mi Mj Q Mj is notQa good candidate for being an upper bound. In [16], 1≤j<i mi−Mj i<j≤c mj−Mi Q Q 3 Römer suggests as upper bound 1 (1.3) β (R/I) ≤ M i j (i−1)!(c−i)! Yj6=i for i = 1,··· ,c and he proved that the lower bound in (1.2) and the upper bound in (1.3) hold if R/I is Cohen-Macaulay of codimension 2 and Gorenstein of codimension 3. It remains open if these bounds hold for other interesting classes of ideals. The goal of this paper is to prove that the lower bound in (1.2) and the upper bound in (1.3) work in the following classes of ideals • standard determinantal ideals of arbitrary codimension c (i.e., ideals generated by the maximal minors of a t×(t+c−1) homogeneous polynomial matrix), • symmetric determinantal ideals defined by the submaximal minors of a t×t homogeneous symmetric matrix, • determinantal ideals defined by the submaximal minors of a t × t homogeneous matrix, and • arithmetically Cohen-Macaulay divisors on a variety of minimal degree. Determinantal ideals are a central topic in both commutative algebra and algebraic geometry. Due to their important role, their study has attracted many researchers and has received considerable attention in the literature. Some of the most remarkable results about determinantal ideals are due to Eagon and Hochster in [3], and to Eagon and Northcottin[4]. EagonandHochster proved thatgenericdeterminantalidealsareperfect. Eagon and Northcott constructed a finite graded free resolution for any determinantal ideal, and as a corollary, they showed that determinantal ideals are perfect. Since then many authors have made important contributions to the study of determinantal ideals, and the reader can look at [2], [1], [12] and [5] for background, history and a list of important papers. Next we outline the structure of the paper. In section 2, we first recall the basic facts on standard determinantal ideals I of codimension c defined by the maximal minors of a t× (t + c − 1) homogeneous matrix A and the associated complexes needed later on. We determine the minimal and maximal shifts in the graded minimal free R-resolution of R/I in terms of the degree matrix U of A and we prove that the lower bound in (1.2) and the upper bound in (1.3) work for standard determinantal ideals of arbitrary codimension c (Theorem 2.4). We also discuss when these bounds are sharp. After some preliminaries, we devote section 3 to prove that the lower bound in (1.2) and the upper bound in (1.3) workfordeterminantal(resp. symmetric determinantal) idealsdefinedbythesubmaximal minors of a t × t homogeneous (resp. symmetric) matrix (Theorems 3.3 and 3.6). We discuss other examples as well as when these bounds are sharp (Theorem 3.8). 2. Standard Determinantal ideals In the first part of this section, we provide the background and basic results on determinantal ideals needed in the sequel, and we refer to [2] and [5] for more details. 4 ROSAM. MIRO´-ROIG Let A be a homogeneous matrix, i.e. a matrix representing a degree 0 morphism φ : F −→ G of free graded R-modules. In this case, we denote by I(A) the ideal of R generated by the maximal minors of A and by I (A) the ideal generated by the j × j j minors of A. Definition 2.1. A homogeneous ideal I ⊂ R of codimension c is called a standard determinantal ideal if I = I(A) for some t×(t+c−1) homogeneous matrix A. Let I ⊂ R be a standard determinantal ideal of codimension c generated by the maximal minors of a t × (t+ c − 1) matrix A = (f )j=1,...,t+c−1 where f ∈ k[x ,...,x ] are ij i=1,...,t ij 1 n homogeneous polynomials of degree a −b . The matrix A defines a degree 0 map j i F = ⊕t R(b ) −A→ G = ⊕t+c−1R(a ) i=1 i j=1 j v 7→ v·A where v = (f ,··· ,f ) ∈ F and we assume without loss of generality that A is minimal; 1 t i.e., f = 0 for all i,j with b = a . If we let u = a − b for all j = 1,...,t + c − 1 ij i j i,j j i and i = 1,...,t, the matrix U = (u )j=1,...,t+c−1 is called the degree matrix associated i,j i=1,...,t to I. By re-ordering degrees, if necessary, we may also assume that b ≥ ... ≥ b and 1 t a ≤ a ≤ ... ≤ a . In particular, we have: 1 2 t+c−1 (2.1) u ≤ u and u ≤ u for all i,j. i,j i+1,j i,j i,j+1 Note that the degree matrix U is completely determined by u , u , ... , u , u , u , 1,1 1,2 1,c 2,2 2,3 ... ,u , ..., u ,u ,... ,u becauseoftheidentityu +u −u −u = 2,c+1 t,t t,t+1 t,c+t−1 i,j i+1,j+1 i,j+1 i+1,j 0 for all i,j. Moreover, the graded Betti numbers in the minimal free R-resolution of R/I(A) depend only upon the integers {u }i≤j≤c+i−1 ⊂ {u }j=1,...,t+c−1 i,j 1≤i≤t i,j i=1,...t as described below. Proposition 2.2. Let I ⊂ R be a standard determinantal ideal of codimension c with degree matrix U = (u )j=1,...,t+c−1 as above. Then we have: i,j i=1,...,t (1) m = u +u +···+u +u +u +···+u for 1 ≤ i ≤ c, i 1,1 1,2 1,i 2,i+1 3,i+2 t,t+i−1 (2) M = u +u +···+u +u +u +···+u for i 1,c−i+1 2,c−i+2 t,t+c−i t,t+c−i+1 t,t+c−i+2 t,t+c−1 1 ≤ i ≤ c, (3) β (R/I) = t+c−1 t+i−2 for i = 1,··· ,c. i t+i−1 i−1 (cid:0) (cid:1)(cid:0) (cid:1) Proof. We denote by ϕ : F −→ G the morphism of free graded R-modules of rank t and t+ c− 1, defined by the homogeneous matrix A associated to I. The Eagon-Northcott complex D (ϕ∗) : 0 0 −→ ∧t+c−1G∗ ⊗S (F)⊗∧tF −→ ∧t+c−2G∗ ⊗S (F)⊗∧tF −→ ... −→ c−1 c−2 ∧tG∗ ⊗S (F)⊗∧tF −→ R −→ R/I −→ 0 0 5 givesusagradedminimalfreeR-resolutionofR/I (See, forinstance[2]; Theorem2.20and [5]; Corollary A2.12 and Corollary A2.13). Now the result follows after a straightforward computation taking into account that t ∧tF = R( b ), i Xi=1 a S (F) = R( b ), a ij 1≤i1≤M···≤ia≤t Xj=1 t+c−1 G∗ = R(−a ) and j Mj=1 b ∧bG∗ = R(− a ). ij 1≤i1<··M·<ib≤t+c−1 Xj=1 (cid:3) Remark 2.3. Let I ⊂ R be a standard determinantal ideal. It is worthwhile to point out that the i-th total Betti number β (R/I) in the minimal free R-resolution of R/I depend i only upon the size t×(t+c−1) of the homogeneous matrix A associated to I. We are now ready to state the main result of this short note. Theorem 2.4. Let I ⊂ R be a standard determinantal ideal of codimension c. Then, we have: m m 1 j j (2.2) ≤ β (R/I) ≤ M i j M −m M −m (i−1)!(c−i)! 1≤Yj<i i j i<Yj≤c j i Yj6=i for i = 1,··· ,c. In addition, the bounds are reached for all i if and only if R/I has a pure resolution if and only if u = u for all 1 ≤ i,r ≤ t and 1 ≤ j,s ≤ t+c−1. i,j r,s Proof. We will first prove the result for J being J ⊂ R a standard determinantal ideal of codimension c whose associated matrix A is a t× (t+c −1) matrix with all its entries linear forms. In this case, for all i = 1,··· ,c, we have (see Proposition 2.2): t+c−1 t+i−2 m (J) = M (J) = t+i−1 and β (R/J) = . i i i (cid:18)t+i−1(cid:19)(cid:18) i−1 (cid:19) 6 ROSAM. MIRO´-ROIG Therefore, R/J has a pure resolution and it follows from [9] and [10] that m (J) m (J) t+j −1 t+j −1 j j = M (J)−m (J) M (J)−m (J) i−j j −i 1≤Yj<i i j i<Yj≤c j i 1≤Yj<i i<Yj≤c = β (R/J) i t+j −1 t+j −1 = i−j j −i 1≤Yj<i i<Yj≤c M (J) M (J) j j = m (J)−M (J) m (J)−M (J) 1≤Yj<i i j i<Yj≤c j i 1 = M (J). j (i−1)!(c−i)! Y j6=i We will now prove the general case. Let I be a standard determinantal ideal of codimension c with associated degree matrix U = (u )j=1,...,t+c−1. Since, for all i = 1,··· ,c, i,j i=1,...,t we have M (I) ≥ m (I) ≥ t+i−1 = m (J) = M (J), i i i i it follows from Proposition 2.2 (3) and Remark 2.3 that β (R/I) = β (R/J) i i t+j −1 t+j −1 = i−j j −i 1≤Yj<i i<Yj≤c 1 = M (J) j (i−1)!(c−i)! Yj6=i 1 ≤ M (I) j (i−1)!(c−i)! Y j6=i and this completes the proof of the upper bound. Let us now prove the lower bound. Recall that if r ≥ m and s ≥ n then u ≥ u . r,s m,n Therefore, for 1 ≤ j < i, we get, using Proposition 2.2, that m = u +u +···+u +u +u +···+u j 1,1 1,2 1,j 2,j+1 3,j+2 t,t+j−1 ≤ (t+j −1)u , t,t+j−1 M −m = u +u +···+u +u +u +···+u i j 1,c−i+1 2,c−i+2 t,t+c−i t,t+c−i+1 t,t+c−i+2 t,t+c−1 − (u +u +···+u +u +u +···+u ) 1,1 1,2 1,j 2,j+1 3,j+2 t,t+j−1 ≥ u +u +···+u t,t+c−i+j t,t+c−i+j+1 t,t+c−1 ≥ (i−j)u , t,t+c−i+j and m (t+j −1)u j t,t+j−1 ≤ M −m (i−j)u i j t,t+c−i+j t+j −1 ≤ . i−j 7 For i < j ≤ c, we have m = u +u +···+u +u +u +···+u j 1,1 1,2 1,j 2,j+1 3,j+2 t,t+j−1 u +u +···+u +···+u +u +···+u , if t+i ≤ j, = 1,1 1,2 1,t+i−1 1,j 2,j+1 t,t+j−1 (cid:26) u1,1 +···+u1,j +u2,j+1+···+ut+i−j,t+i−1 +···+ut,t+j−1, if t+i > j; M −m = u +u +···+u +u +u +···+u j i 1,c−j+1 2,c−j+2 t,t+c−j t,t+c−j+1 t,t+c−j+2 t,t+c−1 − (u +u +···+u +u +u +···+u ) 1,1 1,2 1,i 2,i+1 3,i+2 t,t+i−1 ≥ u +u +···+u t,t+c+i−j t,t+c+i−j+1 t,t+c−1 u +···+u +u +···+u , if t+i ≤ j, ≥ 1,t+i 1,j 2,j+1 t,t+j−1 (cid:26) ut+i−j+1,t+i +···+ut,t+j−1, if t+i > j. Therefore, if i < j ≤ c and t+i ≤ j, we get m u +u +···+u j 1,1 1,2 1,t+i−1 ≤ +1 M −m u +···+u +u +···+u j i 1,t+i 1,j 2,j+1 t,t+j−1 (t+i−1)u 1,t+i−1 ≤ +1 (j −i)u 1,t+i t+i−1 ≤ +1 j −i t+j −1 = , j −i and if i < j ≤ c and t+i > j, we get m u +···+u +u +···+u j 1,1 1,j 2,j+1 t+i−j,t+i−1 ≤ +1 M −m u +···+u j i t+i−j+1,t+i t,t+j−1 (t+i−1)u t+i−j,t+i−1 ≤ +1 (j −i)u t+i−j+1,t+i t+i−1 ≤ +1 j −i t+j −1 = . j −i Hence, putting altogether, we obtain m m t+j −1 t+j −1 j j ≤ M −m M −m i−j j −i 1≤Yj<i i j i<Yj≤c j i 1≤Yj<i i<Yj≤c = β (R/J) i = β (R/I) i and this completes the proof of the lower bound. Checking the inequalities we easily see that we have equality above and below for all 1 ≤ i ≤ c if and only if R/I has a pure (cid:3) resolution. This concludes the proof of the Theorem. Remark 2.5. Since a complete intersection ideal I of arbitrary codimension and Cohen- Macaulay ideals of codimension 2 are examples of standard determinantal ideals, we recover [16]; Theorem 2.1 and Corollary 4.2. 8 ROSAM. MIRO´-ROIG Since the power Is of a complete intersection ideal I ⊂ R is an example of standard determinantal ideal, as a corollary of Theorem 2.4, we have Corollary 2.6. Let I ⊂ R be a complete intersection ideal of codimension c and let s be any positive integer. Then, it holds m m 1 (2.3) j j ≤ β (R/Is) ≤ M i j M −m M −m (i−1)!(c−i)! 1≤Yj<i i j i<Yj≤c j i Yj6=i for i = 1,··· ,c. 3. Ideals defined by submaximal minors The first goal of this section is to prove that the lower bound in (1.2) and the upper bound in (1.3) work for k-algebras k[x ,··· ,x ]/I being I a perfect ideal generated by the 1 n submaximal minors of a t×t homogeneous symmetric matrix. A classical homogeneous ideal that can be generated by the submaximal minors of a t×t homogeneous symmetric matrix is the ideal of the Veronese surface X ⊂ P5. Indeed, the ideal of the Veronese surface X ⊂ P5 = Proj(k[x ,··· ,x ]) can be generated by the 2 × 2 minors of the 1 6 symmetric matrix x x x 6 1 2 x x x . 1 3 4 x x x 2 4 5   Let us now fix some notation. Let I ⊂ S = k[x ,··· ,x ] be a codimension 3, perfect 1 n ideal generated by the submaximal minors of a t × t homogeneous symmetric matrix A = (f ) where f ∈ k[x ,...,x ] are homogeneous polynomials of degree a +a , ji i,j=1,...,t ji 1 n i j i.e. I = I (A). We denote by t−1 2a a +a ··· a +a 1 1 2 1 t a +a 2a ··· a +a  1 2 2 2 t U = . . . . . . . . .   a +a a +a ··· 2a   1 t 2 t t  the degree matrix of A. We assume that a ≤ a ≤ ··· ≤ a . The determinant of A is 1 2 t a homogeneous polynomial of degree ℓ := 2(a + a + ···+ a ). Note that a +a is an 1 2 t i j integer for all 1 ≤ i ≤ j ≤ t while a does not necessarily need to be an integer. i Note that the degree matrix U is completely determined by a , ... , a . Moreover, the 1 t graded Betti numbers in the minimal free S-resolution of S/I (A) depend only upon t−1 the integers a , ... , a as we will describe now. To this end, we recall Jozefiak’s result 1 t about the resolution of ideals generated by minors of a symmetric matrix. LetR bea commutative ringwithidentity andletX = (x )beasymmetric t×tmatrix ij with entries in R. Write Y = (y ) for the matrix of cofactors of X, i.e., y = (−1)i+1Xi ij ij j where Xi stands for the minor of X obtained by deleting the i-th column and the j-th j row of X. The matrix Y is also a symmetric matrix. Let M (R) be the free R-module of t all t×t matrices over R and let A (R) be the free R-submodule of M (R) consisting of t t 9 all alternating matrices. Denote by tr : M (R) −→ R the trace map. By [11]; Theorem t 3.1, the free complex of length 3 associated to X: 0 −→ A (R)−d→3 Ker(M (R)−t→r R)−d→2 M (R)/A (R)−d→1 R t t t t where the corresponding differentials are defined as follows: d (A) = AX, 3 d (N) = XN mod A (R), and 2 t d (Mmod A (R)) = tr(YM) 1 t is acyclic and gives a free resolution of R/I (X). So, we obtain t−1 Proposition 3.1. Let I ⊂ S = k[x ,··· ,x ] be a perfect ideal of codimension 3 generated 1 n by the submaximal minors of a symmetric matrix A. Let 2a a +a ··· a +a 1 1 2 1 t a +a 2a ··· a +a  1 2 2 2 t U = . . . . . . . . .   a +a a +a ··· 2a   1 t 2 t t  be the degree matrix and ℓ := 2(a +a +···+a ). Then, we have: 1 2 t (1) m = ℓ−2a and M = ℓ−2a , 1 t 1 1 (2) m = ℓ−a +a and M = ℓ−a +a , 2 t 1 2 1 t (3) m = ℓ+a +a and M = ℓ+a +a , and 3 1 2 3 t−1 t (4) β (R/I) = t+1 , β (R/I) = t2 −1, and β (R/I) = t . 1 2 2 3 2 (cid:0) (cid:1) (cid:0) (cid:1) Proof. By [11]; Theorem 3.1, I has a minimal free S-resolution of the following type: (3.1) 0 −→ ⊕ S(−a −a −ℓ) −→ (⊕ S(−ℓ−a +a ))/S(−ℓ) −→ 1≤i<j≤t i j 1≤i,j≤t i j ⊕ S(a +a −ℓ) −→ I −→ 0. 1≤i≤j≤t i t So, the maximum and minimum degree shifts at the i-th step are (1) m = ℓ−2a and M = ℓ−2a , 1 t 1 1 (2) m = ℓ−a +a and M = ℓ−a +a , 2 t 1 2 1 t (3) m = ℓ+a +a and M = ℓ+a +a . 3 1 2 3 t−1 t and the i-the total Betti numbers are: (4) β (R/I) = t+1 , β (R/I) = t2 −1, and β (R/I) = t 1 2 2 3 2 which proves what(cid:0)we(cid:1)want. (cid:0) (cid:1) (cid:3) Remark 3.2. Let I ⊂ R bea perfect idealof codimension 3generated by thesubmaximal minors of a symmetric matrix. It is worthwhile to point out that the i-th total Betti number β (R/I) in the minimal free R-resolution of R/I depend only upon the size t×t i of the homogeneous symmetric matrix A associated to I. 10 ROSAM. MIRO´-ROIG We are now ready to bound the i-the total Betti number of codimension 3, perfect ideals generated by the submaximal minors of a symmetric matrix in terms of the shifts in its minimal free R-resolution. Theorem 3.3. Let I ⊂ S be a perfect ideal of codimension 3 generated by the submaximal minors of a t×t symmetric matrix. Then, we have: m m 1 j j (3.2) ≤ β (R/I) ≤ M i j M −m M −m (i−1)!(c−i)! 1≤Yj<i i j i<Yj≤3 j i Yj6=i for 1 ≤ i ≤ 3. In addition, the bounds are reached for all i if and only if R/I has a pure resolution if and only if a = ··· = a . 1 t Proof. We will first prove the result for J being J ⊂ R a codimension 3 perfect ideal generated by the submaximal minors of a t×t symmetric matrix A with linear entries. In this case, for all 1 ≤ i ≤ 3, we have (see Proposition 3.1) m (J) = M (J) = t+i−2, i i t+1 t β (R/J) = , β (R/J) = t2 −1 and β (R/J) = . 1 2 3 (cid:18) 2 (cid:19) (cid:18)2(cid:19) Therefore, R/J has a pure resolution and it follows from [9] and [10] that m (J) m (J) t+j −2 t+j −2 j j = M (J)−m (J) M (J)−m (J) i−j j −i 1≤Yj<i i j i<Yj≤3 j i 1≤Yj<i i<Yj≤3 = β (R/J) i t+j −2 t+j −2 = i−j j −i 1≤Yj<i i<Yj≤3 M (J) M (J) j j = m (J)−M (J) m (J)−M (J) 1≤Yj<i i j i<Yj≤3 j i 1 = M (J). j (i−1)!(3−i)! Yj6=i We will now prove the general case. Let I be a perfect ideal of codimension 3 generated by the submaximal minors of a t×t symmetric matrix, let 2a a +a ··· a +a 1 1 2 1 t a +a 2a ··· a +a  1 2 2 2 t U = . . . . . . . . .   a +a a +a ··· 2a   1 t 2 t t  be its degree matrix and ℓ := 2(a +a +···+a ). Since, for all 1 ≤ i ≤ 3, we have 1 2 t M (I) ≥ m (I) ≥ t+i−2 = m (J) = M (J), i i i i