LEADING COEFFICIENTS AND THE MULTIPLICITY OF KNOWN ROOTS

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1 LEADING COEFFICIENTS AND THE MULTIPLICITY OF KNOWN ROOTS GREGORY J. CLARK AND JOSHUA N. COOPER Abstract. We show that a onic univariate polynoial over a field of characteristic zero, with k distinct non-zero known roots, is deterined by its k proper leading coefficients via an explicit algorith for coputing the ultiplicities of each root. We provide a version of the result and accopanying algorith when the field is not algebraically closed by considering the inial polynoials of the roots. Furtherore, we show how to perfor the aforeentioned algorith in a nuerically stable anner over C, and then apply it to obtain new characteristic polynoials of hypergraphs. 1. Introduction In [7], Győry et al. present the following natural question. Proble 1.1. Let K be a field of characteristic zero. Is it true that a onic polynoial p = n i=0 c ix n i K[x] of degree n with exactly k distinct zeros is deterined up to finitely any possibilities by any k of its non-zero coefficients? By degrees-of-freedo considerations, at least k coefficients are needed; which sets of k coefficients actually suffice, however, sees to be a delicate atter. We consider the following variation of Proble 1.1. The codegree of a onoial ter Cx k in a univariate polynoial f(x) is deg(f) k. A set of coefficients is leading if it corresponds to codegrees {0,..., k} for soe k; it is proper leading if it corresponds to codegrees {1,..., k} for soe k. Proble 1.2. Let K be a field of characteristic zero. Is it true that a onic polynoial p = n i=0 c ix n i K[x] of unknown degree n, with exactly k distinct known zeros r 1, r 2,..., r k, is uniquely deterined by its first k proper leading coefficients? We answer Proble 1.2 in the affirative with the following result. Theore 1.3. Let p = n i=0 c ix n i K[x] be a onic polynoial with k + 1 distinct roots, r 0 = 0, r 1, r 2,..., r k, with ultiplicity 0, 1, 2,... k, respectively. Then the ultiplicities are uniquely deterined by c 0 = 1, c 1,..., c k. Furtherore, p ay be deterined by fewer than k proper coefficients when K is not algebraically closed. Date: June 11, Matheatics Subject Classification. Priary 12E05; Secondary 05C50, 05C65, 12Y05. Key words and phrases. Polynoial root ultiplicity, proper leading coefficients, stable Vanderonde inversion, spectral hypergraph theory. This work was partially supported by a SPARC Graduate Research Grant fro the Office of the Vice President for Research at the University of South Carolina. 1

2 2 GREGORY J. CLARK AND JOSHUA N. COOPER Theore 1.4. Let p = n i=0 c ix n i K[x] be a onic polynoial such that p(0) 0. Suppose p = t i=1 qi i for q i K[x]. The ultiplicity vector = 1,..., t T is uniquely deterined by the t proper leading coefficients if and only if V K t t is non-singular where V i,j = r j. r:q i(r)=0 Reark 1.5. Observe that when q i = x r i (i.e., p splits over K) Theore 1.4 provides the sae conclusion as Theore 1.3. In Section 2 we prove both of the ain results. In particular, we prove Theore 1.3 via an algorith which allows us to copute exactly the ultiplicity of each root. In Section 3 we prove that this algorith is nuerically stable in the sense that the requisite nuber of bits of precision to approxiate each root in order to copute its ultiplicity exactly is linear in k and the logariths of (a) the ratio between the largest and sallest difference of roots, (b) the largest root, and (c) the largest coefficient of codegree at ost k. We conclude by deonstrating the utility of this algorith by coputing previously unknown characteristic polynoials of two 3-unifor hypergraphs in Section Proof of Main Results We begin with a proof of Theore 1.3. Proof of Theore 1.3. Fix such a onic polynoial p with distinct roots r 0 = 0, r 1,..., r k with respective ultiplicities 0, 1,..., k. Ignoring r 0 for a oent, let r = r 1,..., r k T and = 1,..., k T. We denote the Vanderonde atrix and consider Let p K n where V = V 0 = r 1 r 2... r k r1 2 r rk r1 k 1 r2 k 1... r k 1 k r 1 r 2... r k r1 2 r rk r1 k r2 k... rn k p i := k rj i j j=1 then V 0 = p. Notice that V 0 = V diag(r) is non-singular as it is the product of two non-singular atrices. We have then (2.1) = V 1 0 p. We present a forula for p which is a function of only the leading k + 1 coefficients of p..

3 LEADING COEFFICIENTS AND THE MULTIPLICITY OF KNOWN ROOTS 3 Let A be the diagonal atrix where r i occurs i ties and note p(x) = det(xi A). By the Faddeev-LeVerrier algorith (aka the Method of Faddeev, [8]) we have for j 1 (2.2) c j = 1 j c j i tr(a i ) = 1 j c j i p i. j j Let c = c 1, c 2,..., c k T, Λ = diag(1, 1/2,..., 1/k), and c 0 = c 1 c C = c k 1 c k 2... c 0 i=1 By Equation 2.2, c = ΛCp. Moreover, as Λ and C are invertible we have p = (ΛC) 1 c. It follows fro Equation 2.1 that (2.3) = (ΛCV 0 ) 1 c. Furtherore, 0 = n 1. We briefly reark about the proof of Theore 1.3. Proble 1.1 has a flavor of polynoial interpolation: given k points, how any (univariate) polynoials of degree n go through each of the k points? If n k 1 the polynoial is known to be unique and is relatively expensive to copute (as any standard text in nuerical analysis will attest). Our proof technique iics this approach as the classical proble of deterining p = k 1 i=0 c ix (k 1) i, which resebles k distinct points {(x i, y i )} k i=1, can be solved by coputing V T c = y where c = c k 1, c k 2,..., c 0 T, V is as previously defined given r i = x i, and y = y 1,..., y k T. Suppose for a oent that each root is distinct so that k p(x) = (x r i ). i=1 Then c j, the codegree j coefficient, is precisely the jth eleentary syetric polynoial in the variables r 1,..., r k. In the case of repeated roots we have that c j can be expressed using odified syetric polynoials in the distinct roots where r i is replaced with ( i ) j r j i. The expression for each coefficient via these odified syetric polynoials is given by Equation 2.2. Note that if the roots of p are known, it is possible to deterine p with fewer coefficients than the nuber of distinct roots (e.g., when p is a non-linear inial polynoial). We odify Theore 1.3 to include the case when soe of the roots are known to occur with the sae ultiplicity. We now prove Theore 1.4. Proof of Theore 1.4. The proof follows siilarly to that of Theore 1.3. First suppose that V is non-singular. Let = 1,..., t T and t p i = r i j = (V ) i j=1 r:q j(r)=0 i=1

4 4 GREGORY J. CLARK AND JOSHUA N. COOPER so that if V is non-singular, = V 1 p. Let A be defined as in Theore 1.3: the diagonal atrix where the roots of q i occur i ties and note p(x) = det(xi A). We have by the the Faddeev-LeVerrier algorith, for j 1 c j = 1 j c j i tr(a i ) = 1 j c j i p i j j i=1 so that for c = c 1,..., c t T, Λ = diag(1, 1/2,..., 1/t) we have = (ΛCV ) 1 c. If instead is uniquely deterined by the first t proper coefficients then V = (ΛC) 1 c has exactly one solution, hence V is non-singular. As a non-exaple, consider the inial polynoial of α = 2, q α (x) = x 2 2 Z[x] and suppose i=1 p = q d α = x 2d + 0x 2d 1 2dx 2d 2 + Z[x]. Observe that we cannot deterine d given c 1 = 0; oreover, this conclusion is unsurprising, given the hypotheses fro Theore 1.4, since V = [ 2 2] = [0] C 1,1 is singular. However, by inspection we could deterine d given c 2 = 2d and in fact the atrix [ ( 2) 2 ] = [4] C 1 1 is non-singular. Indeed we could deterine d with a siple change of variable: apply Theore 1.4 to p 0 = (y 2) d where y = x The stability of coputing ultiplicities We now consider the feasibility of coputing. In general, the atrix V 0 in Theore 1.3 ay be poorly conditioned, so this calculation is often difficult to carry out even for odest values of k. The goal of this section is to show that if each root of a onic polynoial p(x) Z[x] is approxiated by a disk of radius at ost ɛ, a reasonable precision, then the interval approxiating i, resulting fro a particular algorith, contains exactly one integer. That is, we provide an algorith for exactly coputing via [18] with substantially iproved nuerical stability over siply following the calculations in Section 1. Theore 3.1. Let p(x) = t i=0 c ix t i Z[x] be a onic polynoial with distinct non-zero roots r 1,..., r n such that r 1 r 2 r n > 0. If each root is approxiated by a disk of radius ɛ such that where ɛ < 2 r 2 2n+7 n 5 ( MRc ) n = ( MRc ) n(1+o(1)) M = ax{ax{ r i r j }, 1} and = in{in{ r i r j : i j}, 1} R = ax{ r 1, 1} and r = in{ r n, 1} c = ax{ax{ c i } n i=1, 1}. then the resulting disk approxiating i = (ΛCV 0 ) 1 c i Z contains exactly one integer (i.e., the coputation of is stable).

5 LEADING COEFFICIENTS AND THE MULTIPLICITY OF KNOWN ROOTS 5 Notice that MRc 1 and MRc = 2 for x n 1 when n is even. Roots of unity occur frequently in the spectru of hypergraphs; see Section 3. In particular, k- cylinders essentially k-colorable k-graphs have a spectru which is invariant under ultiplication by the kth roots of unity. Consider now p(x) = x n 1. We have = 2 2 cos( 2π n ) so that n+2 )) 2 ɛ < (2 2 cos( 2π n 2 3n+7 n 5. While ɛ ay see sall, we are chiefly concerned with the nuber of bits of precision needed to approxiate each root. Indeed for x n 1 we need ln ɛ = O(n ln n) bits of precision by the sall-angle approxiation. Reark 3.2. The bound on ɛ is reasonable, as the nuber of bits required to approxiate each root is proportional to the nuber of distinct roots of p and the logariths of the ratio of the sallest difference of the roots with the largest difference of roots, the largest root, and the largest coefficient. In practice, the difficulty of coputing as described in Theore 1.3 is in coputing the inverse of the Vanderonde atrix, whose entries ay vary widely in agnitude and which ay be very poorly conditioned. The task of inverting Vanderonde atrices has been studied extensively. In [4], Eisenberg and Fedele provide a brief history of the topic as well results concerning the accuracy and effectiveness of several known algoriths. However, these algoriths provide good approxiations for the entries of V 1, whereas we seek to express the exactly as eleents of the field of algebraic coplex nubers, since is a vector of integers. In [5], Soto-Eguibar and Moya-Cessa showed that V 1 = W L where is the diagonal atrix 1 r i r k { n k=1,k i : i = j i,j = 0 : i j, W is the lower triangular atrix { 0 : i > j W i,j = n k=j+1,k i (r i r k ) : otherwise, and L is the upper triangular atrix 0 : i < j L i,j = 1 : i = j L i 1,j 1 L i 1,j 1 r i 1 : i [2, n], j [2, i 1]. Using this decoposition, it is possible to copute exactly. To prove Theore 3.1 we first provide an upper bound for the diaeter of the disk approxiating an entry of, W, and L, respectively; to do so, we extensively eploy coputations of [14] found in Chapter 1.3. We present the necessary background here. Let D(z, ɛ) be the open disk in the coplex plane centered at z of radius ɛ. For A = D(a, r 1 ), B = D(b, r 2 ) coplex open disks, we have (1) A ± B = D(a ( ± b, r 1 + r 2 ) (2) 1/B = D, b b 2 r 2 2 r 2 b 2 r 2 2 (3) AB = D(ab, a r 2 + b r 2 + r 1 r 2 )

6 6 GREGORY J. CLARK AND JOSHUA N. COOPER In particular, for the special case of A n we have (3.1) D(a, r 1 ) n = D(a n, ( a r 1 ) n a n ). Moreover, given 0 < r 1 < 1 a (3.2) ( a r 1 ) n a n r 1 (2 a ) n since ( a r 1 ) n a n n k=1 ( ) n r k k 1 a n k r 1 (2 a ) n. Finally, let d(a) = 2r 1 denote the diaeter of A and let A = a + r 1 be the absolute value of A. Then for u C we have (1) d(a ± B) = d(a) + d(b) (2) d(ua) = u d(a) (3) d(ab) B d(a) + A d(b) For the reainder of this paper soe nubers will be exact (e.g., rational nubers) while others will be approxiated by a disk. The non-exact entries of a atrix M C n n will be referred to as disks; this will be clear fro the proble forulation or derived fro the coputations. With a slight abuse of notation we use d(m i,j ) and M i,j to denote the diaeter and absolute value of the disk approxiating the entry M i,j. Moreover, we write d(m) = ax{d(m i,j ) : i, j [n]} and M = ax{ M i,j : i, j [n]}. In the case when the entry is exact, the diaeter is zero and the absolute value (of the disk) is siply the odulus. Theore 3.3. Assue the notations of Theore 1.4, let V denote the Vanderonde atrix fro the proof of Theore 1.3, and let V 1 = W L by [5]. Then ( ) n d(v 1 ) 22n+4 n MR 2 ɛ. and V 1 2n ( ) n RM. Proof. Let D i := D(r i, ɛ) denote the disk centered at r i with radius ɛ. By Equation 3.2 we have for s t (( ) n ) 1 d( ) d D s D t ( ) ( ) n 2 n 2ɛ 2 (2ɛ) 2 2 (2ɛ) 2 since ɛ < /4, and 22n+2 ɛ, n+2 d(w ) d((d s D t ) n ) 2 n+1 M n ɛ, d(l) d(d n s ) (2R) n ɛ.

7 LEADING COEFFICIENTS AND THE MULTIPLICITY OF KNOWN ROOTS 7 We first consider d( W ). Observe that W is upper triangular and each non-zero entry of W is a product of exactly one non-zero entry of and W. In this way ( ) n d(( W ) i,j ) W i,j d( i,i ) + i,i d(w i,j ) 22n+3 M 2 ɛ and ( ) n M W 2. We now deterine d( W L) by first coputing Hence and d(( W ) i,k L k,j ) L k,j d( W i,k ) + W i,k d(l k,j ) 22n+4 2 d(v 1 ) = d( W L) ax i,j n k=1 V 1 2n d(( W ) i,k L k,j ) 22n+4 n 2 ( ) n RM. ( ) n RM ɛ. ( ) n RM ɛ In our coputations we are concerned with V 0 = V diag(r) where diag(r) = diag(r 1,..., r n ) so that V 1 0 = diag(r) 1 V 1. The following Corollary is iediate fro the observation that Corollary 3.4. and d(diag(r) 1 ) 2 r. d(v 1 0 ) 22n+6 n 2 r V 1 0 2n r We are now able to prove Theore 3.1. ( ) n MR ɛ ( ) n RM. Proof of Theore 3.1. Recall = V0 1 C 1 Λ 1 c as defined in the proof of Theore 1.3. Fortunately, the reainder of the coputations are straightforward as C 1, Λ 1, and c have integer, and thus exact, entries. As we have C 1 i,j = d(v 1 0 C 1 ) n(nc n 1 ) 22n+6 n 2 r 0 : i < j 1 : i = j i 1 k=1 c i kc 1 k,j : i > j Further, since Λ 1 = diag(1, 2,..., n) we have ( ) n MR ɛ = 22n+6 n 3 2 rc d(v 1 0 C 1 Λ 1 ) n d(v 1 0 C 1 ) = 22n+6 n 4 2 rc ( ) n MRc ɛ ( ) n MRc ɛ

8 8 GREGORY J. CLARK AND JOSHUA N. COOPER and, finally, d(v0 1 C 1 Λ 1 )c) nc d(v0 1 C 1 Λ 1 ) 22n+6 n 5 ( ) n MRc 2 ɛ < 1 r 2. Thus each interval will contain at ost one integer as desired. 4. Application to Hypergraph Spectra For the present authors, Proble 1.2 arose organically in the context of spectral hypergraph theory. In short, the authors were concerned with deterining highdegree polynoials when the roots (without ultiplicity) are known and all but the lowest-codegree coefficients are too costly to copute. We briefly explain the context of spectral hypergraph theory for those interested in the origin of such questions. However, our presentation of the coputations is self-contained: the reader who wishes to see Theore 1.3 applied iediately ay skip the next few paragraphs. For k 2, a k-unifor hypergraph is a pair H = (V, E) where V = [n] is the set of vertices and E ( ) [n] k is the set of edges. It is coon to refer to such hypergraphs as k-graphs when k > 2 and as just graphs when k = 2. We are particularly interested in the coputation of the characteristic polynoial of a unifor hypergraph. The characteristic polynoial of the adjacency atrix of a graph is straightforward to copute; however, the sae cannot be said for hypergraphs. The characteristic polynoial of the (noralized) adjacency hyperatrix A of H, denoted φ H (λ), is the resultant of a faily of E hoogeneous polynoials 1 of degree k 1 in the indeterinate λ; the order k, diension n hyperatrix A R [n]k, whose rows and coluns are indexed by the vertices of H and whose (v 1,..., v k ) entry is 1/(k 1)! ties the indicator of the event that {v 1,..., v k } is an edge of H is also soeties called the adjacency tensor of H. Equivalently, one can define the characteristic polynoial to be the hyperdeterinant of A λi (as in [9]), where I is the identity hyperatrix, i.e., I(v 1,..., v k ) is the indicator of the event that v 1 = = v k. The set σ(h) = {r : φ H (r) = 0} C is the spectru of H and each r σ(h) is an eigenvalue of H. It is known that φ H (λ) is a onic polynoial of degree n(k 1) n 1, and any of the properties of characteristic polynoials of graphs generalize nicely to hypergraphs; we refer the interested reader to [3] and [15] for further exploration of the topic. Given a k-graph H we ai to copute φ H (λ). Unfortunately, the resultant is known to be NP-hard to copute ([11]) despite its utility in several fields of atheatics, perhaps nowhere ore so than coputational algebraic geoetry. Nonetheless, one can attept to iitate classical approaches to coputing characteristic polynoials of ordinary graphs. In particular, Harary [12] (and Sachs [17]) showed that the coefficients of φ G (λ) can be expressed as a certain weighted su of the counts of subgraphs of G. The authors have established an analogous result for 1 Naely, the Lagrangians of the links of all vertices inus λ ties the (k 1)-st power of the corresponding vertices variables, or, equivalently, the coordinates f v of the gradient of the k-for naturally associated with A λi. The (syetric) hyperdeterinant is the unique irreducible onic polynoial in the entries of A whose vanishing corresponds exactly to the existence of nontrivial solutions to the syste {f v = 0} v V (H).

9 LEADING COEFFICIENTS AND THE MULTIPLICITY OF KNOWN ROOTS 9 the coefficients of φ H (λ) [2]. This forula allows one to copute any low codegree coefficients i.e., the coefficients of x d k for k sall and d = deg(φ H ) by a certain linear cobination of subgraph counts in H. Unfortunately, this coputation becoes exponentially harder as the codegree increases, aking coputation of the entire (often extreely high degree) characteristic polynoial ipossible for all but the siplest cases. A ethod of Lu-Man [13], α-noral labelings is an alternative approach that can obtain all eigenvalues with relative efficiency, but it gives no inforation about their ultiplicities. Cobining these two techniques, however, yields a ethod to obtain the full characteristic polynoial: obtain a list of roots, copute a few low-codegree coefficients using subgraph counts, and then deduce the roots ultiplicities. Therefore, we arrive at the following special case of Proble 1.2. Proble 4.1. Let K be a field of characteristic zero. Is it true that a onic polynoial p K[x] of degree n with exactly k distinct, known roots is deterined by its k proper leading coefficients? Returning to our application of Theore 1.3, we can copute φ H (λ) if we know σ(h) and the first σ(h) coefficients (note this includes coefficients which are zero as well as the leading ter). In [13], Lu and Man introduced α-consistent incidence atrices which can be used to find the eigenvalues of H whose corresponding eigenvector has all non-zero entries. These eigenvalues are referred to as totally non-zero eigenvalues and we denote the set of totally non-zero eigenvalues of H as σ + (H). The authors showed in [1] that for k > 2, σ(h) σ + (H) H H where H = (V 0, E 0 ) H if V 0 V and E 0 E (c.f. Cauchy Interlacing Theore when k = 2). Coputing σ(h) by way of σ + (H) involves solving saller ultilinear systes than the one involved in coputing φ H (λ). Generally speaking, σ(h) is considerably saller than the degree of φ H (λ). In practice, this approach has yielded φ H (λ) when other approaches of coputing φ H (λ) via the resultant have failed. We present two exaples deonstrating these coputations. Consider the huingbird hypergraph B = ([13], E) where E = {{1, 2, 3}, {1, 4, 5}, {1, 6, 7}, {2, 8, 9}, {3, 10, 11}, {3, 12, 13}}. We present a drawing of B in Figure 1 where the edges are drawn as shaded in triangles. Note that deg(φ B ) = n(k 1) n 1 = = and, since B is a hypertree (and thus a 3-cylinder), its spectru is invariant under ultiplication by any third root of unity [3]. We copute the inial polynoials of the totally non-zero eigenvalues of φ B via [10], φ B =x 0 (x 9 6x 3 + 8x 3 4) 1 (x 9 5x 6 + 5x 3 2) 2 (x 3 1) 3 (x 6 4x 3 + 2) 4 (x 9 4x 6 + 3x 3 1) 5 (x 6 3x 3 + 1) 6 (x 3 3) 54 (x 3 2) 7.

10 10 GREGORY J. CLARK AND JOSHUA N. COOPER With the intent of applying Theore 1.3 to φ B we consider the change of variable y = x 3 and observe that we need to deterine c 3, c 6,..., c 48 as there are sixteen distinct nonzero cube roots. We copute c 3 = c 6 = c 9 = c 12 = c 15 = c 18 = c 21 = c 24 = c 27 = c 30 = c 33 = c 36 = c 39 = c 42 = c 45 = c 48 = Using Theore 3.1 we have M < 3, >.14, R < 4.39, r >.38, and c = c 48 so that each root of φ B needs to be approxiated to at ost 3091 bits of precision. Using SageMath ([18]), we obtain φ(b) =x (x 9 6x 3 + 8x 3 4) 729 (x 9 5x 6 + 5x 3 2) 972 (x 3 1) 1782 (x 6 4x 3 + 2) 486 (x 9 4x 6 + 3x 3 1) 324 (x 6 3x 3 + 1) 216 (x 3 3) 54 (x 3 2) 119. In Figure 1 we provide a plot of σ(φ B ) drawn in the coplex plane where a disk is centered at each root and each disk s area is proportional to the algebraic ultiplicity of the underlying root in φ B. Now consider the Rowling hypergraph 2 R = ([7], {1, 2, 3}, {{1, 4, 5}, {1, 6, 7}, {2, 5, 6}, {3, 5, 7}}). A drawing of R is given in Figure 2 where the edges are drawn as arcs and its spectru is drawn siilarly to that of φ B ; note that R is also the Fano plane inus two edges. We have deg(φ R ) = n(k 1) n 1 = = 448. It is easy to verify that R is not a 3-cylinder; however, its spectru is invariant under ultiplication by any third root of unity (see Lea 3.11 of [6]). By [13] we 2 The nae was chosen for its reseblance to an iportant narrative device in [16].

11 LEADING COEFFICIENTS AND THE MULTIPLICITY OF KNOWN ROOTS 11 Figure 1. The huingbird hypergraph and its spectru. have φ R =x 0 (x 3 1) 1 (x 15 13x x 9 147x x 3 64) 2 (x 6 x 3 + 2) 3 (x 6 17x ) 4 With the intent of applying Theore 3.3 we need to deterine only c 3, c 6, c 9, c 12. We have c 3 = 240 c 6 = c 9 = c 12 = By Theore 3.1 we have M < 4.5, >.69, R < 2.25, and r = 1 so that at ost 252 digits of precision are required to approxiate each root. We copute φ R =x 133 (x 3 1) 27 (x 15 13x x 9 147x x 3 64) 12 (x 6 x 3 + 2) 6 (x 6 17x ) 3 Figure 2. The Rowling hypergraph and its spectru.

12 12 GREGORY J. CLARK AND JOSHUA N. COOPER 5. Acknowledgents Thanks to Alexander Duncan for helpful discussions and insights. References 1. G. Clark and J. Cooper, On the adjacency spectra of hypertrees, The Electronic Journal of Cobinatorics. To appear. 2. G. Clark and J. Cooper, A cobinatorial description for coefficients of the characteristic polynoial of a hypergraph. In preparation. 3. J. Cooper, A. Dutle, Spectra of unifor hypergraphs, Linear Algebra Appl. 436 (2012) A. Eisinberg, G. Fedele, On the inversion of the Vanderonde atrix, Applied Matheatics and Coputation, 174 (2)(2006), F. Soto-Eguibar, H. Moya-Cessa, Inverse of the Vanderonde and Vanderonde confluent atrices, Applied Matheatics & Inforation Sciences, 5(3)(2011), Y. Fan, T. Huang, Y. Bao, C. Sun, and Y. Li, The Spectral Syetry of Weakly Irreducible Nonnegative Tensors and Connected Hypergraphs. arxiv: K. Győry, L. Hajdu, À. Pintèr, and A. Schinzel, Polynoials Deterined by a few of their Coefficients, Indag. Mathe., N.S., 15 (2), F. R. Gantacher, The Theory of Matrices. NY: Chelsea Publishing, I. M. Gelfand, A. V. Zelevinskii, M. M. Kapranov, Discriinants, Resultants, and Multidiensional Deterinants, Birkhäuser Boston, D. Grayson, and M. Stillan, Macaulay2, a software syste for research in algebraic geoetry, Available at B. Grenet, P. Koiran, N. Portier, The Multivariate Resultant Is NP-hard in Any Characteristic. Matheatical Foundations of Coputer Science, Lecture Notes in Coputer Science, 6281 (2010) F. Harary, The deterinant of the adjacency atrix of a graph, SIAM Rev., 4 (1962), L. Lu and S. Man, Connected hypergraphs with sall spectral radius, Linear Algebra Appl. 509 (2016), M. Petković and L. Petković. Coplex Interval Arithetic and Its Applications. Berlin: Wiley- VCH, L. Qi, Eigenvalues of a real supersyetric tensor, J. Sybolic Coput. 40 (2005), J. K. Rowling, Harry Potter and the Deathly Hallows. New York, NY, H. Sachs, Über Teiler, Faktoren und charakteristiche Polynoe von Graphen, Teil I, 12 (1966), SageMath, the Sage Matheatics Software Syste (Version 7.2), The Sage Developers, 2018, W. Zhang, L. Kang, E. Shan, Y. Bai, The spectra of unifor hypertrees, Linear Algebra Appl. (2017), Departent of Matheatics, University of South Carolina, Colubia, South Carolina E-ail address: gjclark@ath.sc.edu Departent of Matheatics, University of South Carolina, Colubia, South Carolina E-ail address: cooper@ath.sc.edu