On the inverse eigenvalue problems: the case of superstars

Abstract

Abstract. Let T be a tree and let x 0 be a vertex of T . T is called a superstar with central vertex x 0 if T − x 0 is a union of paths. The General Inverse Eigenvalue Problem for certain trees is partially answered. Using this description, some superstars are presented for which the problem of ordered multiplicity lists and the Inverse Eigenvalue Problem are not equivalent.Key words. Eigenvalues, Tree, Graph, Symmetric matrices. AMS subject classifications. 15A18, 05C38, 05C50. Introduction.Let A = [a ij ] be an n × n real symmetric matrix. We denote by G(A) = (X, U ) the simple graph on n vertices, {1, . . . , n}, such that {i, j} ∈ U , i = j, if and only if a ij = 0. Let A(i) denote the principal submatrix of A obtained by deleting row and column i.Let G = (X, U ) a simple graph, where X = {x 1 , . . . , x n } is the vertex set of G and let S(G) be the set of all n × n real symmetric matrices A such that G(A) ∼ = G. One of the most important problems of Spectral Graph Theory is the General Inverse Eigenvalue Problem for S(G) (GIEP for S(G)):"What are all the real numbers λ 1 ≤ . . . ≤ λ n and µ 1 ≤ . . . ≤ µ n−1 that may occur as the eigenvalues of A and A(i), respectively, as A runs over S(G)?" Another important problem is the Inverse Eigenvalue Problem for S(G) (IEP for S(G));"What are all the real numbers λ 1 ≤ . . . ≤ λ n that may occur as the eigenvalues of A, as A runs over S(G)?" First, we remind the reader of some results concerning the GIEP.Perhaps the most well known result on this subject is the Interlacing Theorem If λ is a real number and A is an n × n real symmetric matrix, we denote by m A (λ) the multiplicity of λ as an eigenvalue of A. As a Corollary of Theorem 1.1 we have the following result: Proposition 1.2.[11] Let A be an n × n Hermitian matrix and let λ be an eigenvalue of A. ThenWhen the graph G is a path, we have the well-known fact: The solution of the GIEP for S(G) when G is a cycle is also well known, see [3,4,5,6].Leal Duarte generalized the converse of Proposition 1.3 to any tree, [13]. Proposition 1.4. Let T be a tree on n vertices and let i be a vertex of T . Let λ 1 < . . . < λ n and µ 1 < . . . < µ n−1 be real numbers. Ifthen there exists a matrix A in S(T ), with eigenvalues λ 1 < . . . < λ n , and such that, A(i) has eigenvalues µ 1 < . . . < µ n−1 .In [7], Johnson and Leal Duarte studied this problem for vertices, of a generic path T , of degree two and solved it for the particular case that occurs when A is a matrix in S(T ) and A(i) has eigenvalues of multiplicity two. "Let G be a simple connected graph G on n vertices, x 0 be a vertex of G of degree k and G 1 , . . . , G k be the connected components of G − x 0 . Let λ 1 , . . . , λ n be real numbers, g 1 (t), . . . , g k (t) be monic polynomials having only real roots and such that Let g(t) be a monic polynomial of degree s + 1. There exists a matrix A in S(T ) with characteristic polynomial if and only if the roots of g(t) strictly interlace those ofThe statement of the previous theorem is shorter when T is a general...

The maximum multiplicity and the two largest multiplicities of eigenvalues in a Hermitian matrix whose graph is a tree

“…We refer to such a list of multiplicities as the unordered multiplicity list and we denote it by (m 1 It is well known that M 1 (T) is equal to the path cover number P(T), the smallest number of nonintersecting induced paths of T that cover all the vertices of T; this is the same as max(p − q), where p is the number of paths remaining when q vertices have been removed from T in such a way as to leave only induced paths [3]. Remark 1.2.…”