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A note on “Methods for constructing distance matrices and the inverse eigenvalue problem”

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Abstract

In this paper, a symmetric nonnegative matrix with zero diagonal and given spectrum, where exactly one of the eigenvalues is positive, is constructed. This solves the symmetric nonnegative eigenvalue problem (SNIEP) for such a spectrum. The construction is based on the idea from the paper Hayden, Reams, Wells, “Methods for constructing distance matrices and the inverse eigenvalue problem”. Some results of this paper are enhanced. The construction is applied for the solution of the inverse eigenvalue problem for Euclidean distance matrices, under some assumptions on the eigenvalues.

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... Euclidean distance matrices were introduced by Menger in 1928, later they were studied by Schoenberg [22], Gower [9], and other authors. In recent years many new results were obtained (see [13,15] and the references therein). EDMs have many interesting properties and are used in various applications in linear algebra, graph theory, bioinformatics, e.g., where frequently a question arises, what can be said about a configuration of points x i , if only distances between them are known. ...
... A nonzero EDM has exactly one positive eigenvalue and the sum of its eigenvalues is zero. It is conjectured that there always exists a solution of the inverse eigenvalue problem, i.e., to prove that any set of numbers that meet these conditions is a spectrum of an EDM (see [15,17], e.g). ...
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... In the paper [8], G. Jaklič and J. Modic offered a method for constructing a symmetric nonnegative matrix with zero diagonal and eigenvalues λ i , where n i=1 λ i = 0 and λ 1 > 0 > λ 2 · · · λ n , then they survey the inverse eigenvalue problem for Euclidean distance matrices, which are a subclass of such matrices. Although this method is useful for solving many problem of inverse eigenvalue problems of distance matrices, but it has some constraints, for example large matrix dimension, and there is no solution for some special spectrum. ...
... Theorem 4.7. [11] Let n ≥ ℓ ≥ 2 and let λ 1 ≤ · · · ≤ λ n−ℓ+1 < λ n−ℓ+2 = · · · = λ n−1 = 0 < λ n be real numbers such that n i=1 λ i = 0. Then there is a nonnegative matrix A ∈ S 0 (K ℓ−1 ∨ K n−ℓ+1 ) such that spec(A) = {λ 1 , . . . , λ n−ℓ+1 , 0 (ℓ−2) , λ n }. ...
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A hollow matrix described by a graph $G$ is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in $G$. For a given graph $G$, the determination of all possible spectra of matrices associated with $G$ is the hollow inverse eigenvalue problem for $G$. Solutions to the hollow inverse eigenvalue problems for paths and complete bipartite graphs are presented. Results for related subproblems such as possible ordered multiplicity lists, maximum multiplicity of an eigenvalue, and minimum number of distinct eigenvalues are presented for additional families of graphs.
... For instance, T.L. Hayden, R. Reams and J. Wells have solved the inverse eigenvalue problem for Euclidean distance matrices of order n = 3, 4, 5, 6 , and any n for which there exists a Hadamard matrix and also they solved this problem: If for n ∈ ℕ there exists a Hadamard matrix of order n, then there is an (n + 1) × (n + 1) and an (n + 2) × (n + 2) distance matrix with eigenvalues which hold under special conditions for n ⩽ 16 [13]. In the paper [14], Jaklic̆ and Modic offered a method for constructing a symmetric nonnegative matrix with zero diagonal and eigenvalues i , where ∑ n i=1 i = 0 and 1 > 0 > 2 ⩾ ⋯ ⩾ n , then they survey the inverse eigenvalue problem for Euclidean distance matrices, which are a subclass of such matrices. Nazari and Mahdinasab solved this problem without using any Hadamard matrix [15]. ...
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In this paper, at first, for a given set of real numbers with only one positive number, and in continue for a given set of real numbers in special conditions, we construct a symmetric nonnegative matrix such that the given set is its spectrum.
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In this paper, a relation between graph distance matrices and Euclidean distance matrices (EDM) is considered. Graphs, for which the distance matrix is not an EDM (NEDM-graphs), are studied. All simple connected non-isomorphic graphs on n <= 8 nodes are analysed and a characterization of the smallest NEDM-graphs, i.e., the minimal forbidden subgraphs, is given. It is proven that bipartite graphs and some subdivisions of the smallest NEDM-graphs are NEDM-graphs, too.
... A nonzero EDM has only one positive eigenvalue λ 1 , and the sum of its eigenvalues is zero. It is conjectured that any set of numbers that meet these conditions can be a spectrum of an EDM (see, e.g., [10,11]). ...
Article
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Chapter
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