a general formula for nding the eigenvalues of a n-star (an n-star is a graph with nboundary vertices whose only connection is to a single, central interior vertex). /LastChar 196 The necessary and sufficient conditions may be elegantly formulated by means of the notion of majorization, going back to the early work of Muirhead [20] (see also [12,19]). Then the conditions0<12<2232n12n21, then pi1(N)=pi+1(N)=1(i=2,,rN1);{N11,N21,,Nn11}{p1(N),p2(N),,prNn11(N)};are necessary and sufficient such that there exist a collection of (positive) masses {mk(j)}k=1nj, a mass M>0, and lengths {lk(j)}k=0nj(j=1, 2,,q) with k=0njlk(j)=lj such that the spectral problem (N1) on the corresponding star graph has the set N as Neumann eigenvalues. /Subtype/Type1 /Name/F11 10, 115119 (2021). At the same time, Chapuy and Feray[6] showed that the integrality of the Star graphs was already solved in another context, since it is equivalent to studying the spectrum of JucysMurphy elements in the algebra of the symmetric group[11]. 2.9] used above for the case M=0 also applies in the case M>0. %PDF-1.5 In the following we consider the properties of the set of Neumann eigenvalues alone. 783.4 872.8 823.4 619.8 708.3 654.8 0 0 816.7 682.4 596.2 547.3 470.1 429.5 467 533.2 Since the number of the former is equal to n2 by Lemma 2.6 (ii) and the number of the latter is trivially less than or equal to the number rNn11 of entries of the majorized vector, we obtain n2rNn11. : On the eigenvalues multiplicity function of the Star graph. /FontDescriptor 23 0 R ?o&l#&6 ZC NXj`k{w|#:o.w7ap+ In order to prove (2.14), we note that Theorem 2.7 (2) yields that for the vectors (pj(D))j=1rD, (pj(N))j=1rN of Dirichlet and Neumann multiplicities ordered decreasingly, we have(2.15) pj(D)>1,pj(N)=pj(D)1(j=1,2,,rD),(2.15) (2.16) pj(D)=1,pj(N)=pj(D)(j=rD+1,,rD). 756.4 705.8 763.6 708.3 708.3 708.3 708.3 708.3 649.3 649.3 472.2 472.2 472.2 472.2 Register to receive personalised research and resources by email. 39 0 obj hKAgHKdcjDR4 C ZQDAt_D- For details we refer the reader to [21, Sect. The trace of A is the sum of the eigenvalues of A, each taken with the same multiplicity as it occurs among the roots of the equation det(AI) = 0. 465 322.5 384 636.5 500 277.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 To describe a combinatorial approach for calculating multiplicities of eigenvalues of the Star graphs \(S_n\), \(n\geqslant 2\), we need to give basic definitions and notation on representation theory of the symmetric group[16]. Condition (4) in Remark 3.5 is condition (c) in [13, Thm. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 458.3 458.3 416.7 416.7 We remark that the constructive procedure in [21, Thm. If a vectorx=(xi)i=1smajorizes a vector(yi)i=1t, then the number of non-zero entries ofxis less or equal to the number of non-zero entries ofy,xy#{i{1,,s}:xi>0}#{i{1,,t}:yi>0}.In fact, denote the two numbers bys0, t0and assumes0>t0. Throughout this paper, we consider a plane star graph of q Stieltjes strings, qN, q2, joined at the central vertex called the root where a mass M0 is placed and with all q pendant vertices fixed; here, following Gantmakher and Krein (see [8,23]), a Stieltjes string is a thread (i.e. 26, Odessa 65020, Ukraine, Mathematisches institut, Universitt Bern, Sidlerstr. To this end, in the case M=0, we set n+1:= for convenience. MathSciNet 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Claims (2) and (2) follow from Theorem 2.7 (2) (see [21, Cor. [1] proposed the following . (4.2) Then n=7 andN1=3,N2=3,N3=1,rD=3,rN=5.The sequences {k}k=17 and {k}k=17 are interlacing as required in assumption (1) of Theorem 2.9. We discuss graphs with just two main eigenvalues in the context of measures of irregularity, and in the context of harmonic graphs. /Name/F3 on p. 19]. Siber. 9]. /Name/F8 Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. For an eigenvalue , let E()denote the eigenspace {xR|A(G)x=x}. 531.3 826.4 826.4 826.4 826.4 0 0 826.4 826.4 826.4 1062.5 531.3 531.3 826.4 826.4 Moreover, by the definition of rN as the number of multiple positive elements of N, we havepk(N)>1,pk(D)=pk(N)+1(k=1,2,,rN),pk(N)=1,pk(D)=pk(N)=1(k=rN+1,,rNn1),pk(D)=1(k=rNn1+1,,rD).This shows that the majorized vector on the right hand side of (3.4) is, in fact, equal to (p1(D),,prD(D)) and hence(N1,,Nn1)(p1(D),,prD(D)).Thus we have shown that the sequences N and D satisfy all assumptions of Theorem 3.3 which yields the claim. N:={{k}k=1n+1ifM>0,{k}k=1nifM=0,k=k, 00,{k}k=1nifM=0,k=k, 00. hXiTSgv B"B A@Ej5P+j^(QEelR-*c:{:AH9rsyyM 8 H~ zA;*6KD@5}y2MBo GhrlX/;}fN>ai,K,vF*p" {q\y|O/ur? 743.3 743.3 613.3 306.7 514.4 306.7 511.1 306.7 306.7 511.1 460 460 511.1 460 306.7 It is known that the spectrum of S n is integral, i.e. ]`A Then Theorem 2.7 (1) and (2) show thatk010 the eigenvalue problem (3.5) is equivalent to the spectral problem for the star-patterned matrix 1/21/2. {(j)}=1nj, (j)=(j), 0<(j)<(j) for 0<<, are the distinct eigenvalues of the Dirichlet problem (2.5)(2.7) on the j-th edge for j=1,2,,q. endobj . The same applies for Theorem 3.4 with just one sequence being the Neumann spectrum. (i) we showed that (1) and (2) imply that the number of simple Neumann eigenvalues satisfies N(1)rD+1; on the other hand, rDn1 by Remark 2.2 (ii). << Moreover, \(\pm (n-1)\) are simple eigenvalues of \(S_n\). We use cookies to improve your website experience. [6,16,2]). The aim of the present paper is to study the problem where, in addition to the number of edges and the lengths of all edges, also the number of masses on each edge is prescribed. The complete graph Kn has an adjacency matrix equal to A = J I, where J is the all-1's matrix and I is the identity. Following condition (2), we choose {k}k=1n, as follows: If k{1,2,,n} and k0}#{i{1,,t}:yi>0}.In fact, denote the two numbers bys0, t0and assumes0>t0. /Name/F5 /Subtype/Type1 Example 4.2 Given q=3, n1=3, n2=n3=2, l1=12, l2=l3=1, and(4.1) 12=35,12=1,22=52,22=32=32=42=42=4,(4.1) (4.2) 52=3+5,52=62=62=72=72=6. 460 511.1 306.7 306.7 460 255.6 817.8 562.2 511.1 511.1 460 421.7 408.9 332.2 536.7 Siber. endobj , The necessity of (4), (5), (6) and of (4), (5), (6), respectively, follows in the same way as in the proof of Remark 2.10. endstream endobj startxref 277.8 500] By Theorem 2.7 (1) and (2), the only two possibilities for a simple Neumann eigenvalue k with k{2,,n} to appear is either strictly between two Dirichlet eigenvalues or coinciding with a double Dirichlet eigenvalue, i.e.k10, the simple eigenvalue n+1. 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 If rN denotes the number of distinct positive Neumann eigenvalues and rD the number of positive Dirichlet eigenvalues with multiplicity greater than one, then(2.13) rN={rD+rD+1ifM>0,rD+rDifM=0,(2.13) and(2.14) (2.14). }(n^2-4n+2).\\ \mathrm {mul}(n-4)= & {} \frac{(n-1)(n-2)}{3!} It is clear that an eigenvalue of G is always an eigenvalue of its switchings, but being main is not necessarily preserved. First we show that it is possible to divide the elements of the set {k}k=1n into q groups {(j)}=1nj for j=1,2,,q such that(3.1) D={k}k=1n:=j=1q{(j)}=1nj,(3.1) with (j)=(j) and 0<(j)<(j) for 0<<. Claim (6) is a consequence of the majorization property (3). Denote by rD the number of distinct positive Dirichlet eigenvalues, by p(D)=(pi(D))i=1rD the vector of their multiplicities in decreasing order, and by Nj the number of edges for which the number of masses is j(j=1,2,,n1), i.e. << The results of our computations are summarized in electronic supplementary material. /Filter[/FlateDecode] Then. /FirstChar 33 endobj If we use claim (5) already proved and add +1 in the n1 components of the majorizing vector (N11,N21,,Nn11) in condition (3) and, on the right hand side, add +1 in the first rN components and n1rN new components 1, we conclude that, for M>0,(3.4) (N1,,Nn1)(p1(N)+1,,prN(N)+1,prN+1(N),,prNn11(N),1,,1n1rN);(3.4) note that this is possible since rNrNn1+1, i.e. (2.16) This implies the first and the last equality in (2.14) if we note (2.13). 5 Howick Place | London | SW1P 1WG. The first claim follows from the first inequality N1p1(D) of the majorization property (3) in Theorem 2.7 since N1=q by definition. Electron. endobj Moreover, a lower bound on multiplicity of eigenvalues of \(S_{n}\) for sufficiently large n was obtained. In this paper we observe methods for getting explicit formulas of eigenvalue multiplicities in the Star graphs \(S_n\), present such formulas for the eigenvalues \(\pm (n-k)\), where \(2\leqslant k \leqslant 12\), and finally collect computational results of all eigenvalue multiplicities for \(n\leqslant 50\) in the catalogue. 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 722.6 693.1 833.5 795.8 382.6 2.5] to prove the claim in the case M>0. Again by assumption (3), we conclude that(3.2) (N11,N21,,Nn11)(p1(D)1,,prD(D)1,prD+1(D),,prDn1+rD(D))=(p1(D)1,,prD(D)1,1,,1). This follows sinceN1=7,N2=4,N3=3,N4=2,N5=1,p1(N)=6,p2(N)=4,pj(N)=1,j=3,4,,9,and hence (N11,N21,N31,N41,N51)=(6,3,2,1,0)(6,4,1,1)=(p1(N),p2(N),p3(N),p4(N))because 6+36+4. endobj 511.1 575 1150 575 575 575 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The following elementary lemma on the inversion of the non-increasing function {1,2,,q}N, jnj, will be used throughout this paper. If s=t, then x is said to majorize y, written as xy, if(2.9) :i=1txi=i=1tyi,i=1xii=1yi(=1,2,,t1). Note that conditions (4) and (5) are not comparable and analogously for (4) and (5). Moreover, our method allows to construct the matricesandexplicitly if a set of lengths{lj}j=1qis given (see Example 4.2 below). {:\C?n]""EB+1i}F'x>\6QhmNYkCEg#|o7')Nata5JN}*_~lC' WeP2j5v53[`C+%9_o= Our main results include necessary (Theorem 2.7) and sufficient (Theorem 3.3) conditions for two (finite) sequences {k} and {k}, counted with multiplicities, to be the Dirichlet and Neumann spectra of a star graph of Stieltjes strings as described above, as well as necessary (Theorem 2.9) and sufficient (Theorem 3.4) conditions for one (finite) sequence {k}, counted with multiplicities, to be the Neumann spectrum. 525 768.9 627.2 896.7 743.3 766.7 678.3 766.7 729.4 562.2 715.6 743.3 743.3 998.9 From the early days, rep-resentation theory and number theory have been very useful for examining the spectra of strongly regular graphs with symmetries. INTRODUCTION Let G be a simple graph with vertex set V(G) = {1,2,.,n} and (0,1)-adjacency matrix A. We prove that and 1 are the eigenvalues of the signed complete graph with the multiplicity at least t if there are vertices whose all incident edges are positive or negative, respectively. We establish necessary and sufficient conditions on the location and multiplicities of two (finite) sequences of numbers {k} and {k} to be the corresponding Dirichlet and Neumann eigenvalues. /LastChar 196 While Theorem 3.4 provides necessary and sufficient conditions for a sequence {k2}k=1n+1, counted with multiplicities, to be the eigenvalues of a star-patterned matrix1/21/2, [13, Thm. Then there exist a collection of masses {mk(j)}k=1nj and lengths {lk(j)}k=0nj(j=1,2,,q) with k=0njlk(j)=lj such that the corresponding spectral problems (N1) and (D1) with M=0 have the sets N={k}k=1n and D=j=1qD(j) as Neumann and Dirichlet eigenvalues. Can one distinguish quantum trees from the boundary? /FirstChar 33 k(k,k+1), with multiplicity 1; if one of k, k+1 is multiple, say k+1==k+pj0(N) with multiplicity pj0(N)2 for some j0{1,2,,rN} (whence k0.D(2)+rDifM=0,N(l)=D(l+1)(l=2,3,,q1). In the following we will show that the assumptions above ensure the existence of a sequence D:={k}k=1n, k>0, k:=k for k>0, such that the two sequences N and D satisfy the assumptions of Theorem 3.3. (4)the number of simple Neumann eigenvalues is at least[rN/2]+1, (5)the number of simple Neumann eigenvalues is at leastn1+1,while (3) implies. Subtracting the identity shifts all eigenvalues by 1, because Ax = (J I)x = Jx x. . In the sequel, we label the edges of the star graph by j=1,2,,q such that the j-th edge carries nj>0 point masses mk(j) in its interior (k=1,2,,nj) and n1n2nq; here we do not count the possible mass at the centre and there are no masses at the pendant vertices. We denote a partition of n as \(\lambda \vdash n\). Two representations \(\rho _1:G\rightarrow GL(V_1)\) and \(\rho _2:G\rightarrow GL(V_2)\) are equivalent if there exists a bijective linear map \(\varphi : V_1\rightarrow V_2\) such that \(\varphi \rho _2(g)=\rho _1(g)\varphi \) for all \(g\in G\). Condition (3) in Theorem 3.4 is condition (d) in [13, Thm. 41 0 obj /Subtype/Type1 Due to the majorization assumption (3), together with the property that nj=#{i{1,2,,n1}:Nij} by Lemma 2.6 (iv) and with j=1qnj=n, we may continue like this to obtain q groups of elements {(j)}=1nj (j=1,2,,q) such that (3.1) holds. In this paper, the eigenvalues of signed complete graphs are investigated. If XV(G), denote V(G)Xby X. We assume that this web is stretched and study the small transverse vibrations vk(j)(t) of the masses mk(j) in two different cases (keeping the notation in [21]): the mass M at the central vertex is free to move in the direction orthogonal to the equilibrium position of the strings (Neumann problem). /Widths[306.7 514.4 817.8 769.1 817.8 766.7 306.7 408.9 408.9 511.1 766.7 306.7 357.8 Moreover, by the definition of rN as the number of multiple positive elements of N, we havepk(N)>1,pk(D)=pk(N)+1(k=1,2,,rN),pk(N)=1,pk(D)=pk(N)=1(k=rN+1,,rNn1),pk(D)=1(k=rNn1+1,,rD).This shows that the majorized vector on the right hand side of (3.4) is, in fact, equal to (p1(D),,prD(D)) and hence(N1,,Nn1)(p1(D),,prD(D)).Thus we have shown that the sequences N and D satisfy all assumptions of Theorem 3.3 which yields the claim. Then, 1 t A 1 = $ 1 n j, j A i j = 2 e n, so 1 2 e n (which is the average degree). This shows that N(1)=rD+D(2)+1 if M>0 and N(1)=rD+D(2) if M=0.If k is an eigenvalue of multiplicity equal to l2, then k1 and, if M>0, also kn+1 and there exists a k0{2,,nl} such that k0k and k01> Moreover, by [21, Thm. 572 0 obj <>/Filter/FlateDecode/ID[<0A159915081B1F48AF39AE2466EFE3AB>]/Index[313 348]/Info 312 0 R/Length 602/Prev 457355/Root 314 0 R/Size 661/Type/XRef/W[1 3 1]>>stream Explicitly, a star with edges (and therefore, vertices) is a graph where one of the vertices is adjacent to all other vertices, and there are no edges other than the edges involving that vertex.

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