The distance matrix of a graph G is the matrix containing the pairwise distances between vertices. The distance eigenvalues of G are the eigenvalues of its distance matrix and they form the distance spectrum of G. We determine the distance spectra of halved cubes, double odd graphs, and Doob graphs, completing the determination of distance spectra of distance regular graphs having exactly one positive distance eigenvalue. We characterize strongly regular graphs having more positive than negative distance eigenvalues. We give examples of graphs with few distinct distance eigenvalues but lacking regularity properties. We also determine the determinant and inertia of the distance matrices of lollipop and barbell graphs.

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The distance matrix of a connected graph is the symmetric matrix with columns and rows indexed by the vertices and entries that are the pairwise distances between the corresponding vertices. We give a construction for graphs which differ in their edge counts yet are cospectral with respect to the distance matrix. Further, we identify a subgraph switching behavior which constructs additional distance cospectral graphs. The proofs for both constructions rely on a perturbation of (most of) the distance eigenvectors of one graph to yield the distance eigenvectors of the other.

We give a construction of a family of (weighted) graphs that are pairwise cospectral with respect to the normalized Laplacian matrix, or equivalently probability transition matrix. This construction can be used to form pairs of cospectral graphs with differing number of edges, including situations where one graph is a subgraph of the other. The method used to demonstrate cospectrality is by showing the characteristic polynomials are equal.

To approximate solutions of a linear differential equation, we project, via trigonometric interpolation, its solution space onto a finite-dimensional space of trigonometric polynomials and construct a matrix representation of the differential operator associated with the equation. We compute the ranks of the matrix representations of a certain class of linear differential operators. Our numerical tests show high accuracy and fast convergence of the method applied to several boundary and eigenvalue problems.MSC 34K28, 65N22, 15A03, 34B05, 34B09, PACS 0002 IntroductionIn this paper, we use trigonometric interpolation to approximate solutions of a differential equation Au = f , whose differential operator A with domain D(A) is a formal polynomial of operators 1, x, d dx , and f ∈ Range(A). A solution u is projected onto the space T L n of L-periodic trigonometric polynomials of degree n. The projection T [u], defined as a trigonometric interpolant of u, is identified with a vectorû of its values at N = 2n+1 partition points via an isomorphism ψ, where n ∈ N. The operator A is represented by an N × N matrix A defined implicitly by Av = ψT [AT [v]] for all v ∈ D(A). The original equation is approximated by a system of linear equations Aû =f , wheref = ψT [f ].