15,462 research outputs found
On a Problem of Harary and Schwenk on Graphs with Distinct Eigenvalues
Harary and Schwenk posed the problem forty years ago: Which graphs have
distinct adjacency eigenvalues? In this paper, we obtain a necessary and
sufficient condition for an Hermitian matrix with simple spectral radius and
distinct eigenvalues. As its application, we give an algebraic characterization
to the Harary-Schwenk's problem. As an extension of their problem, we also
obtain a necessary and sufficient condition for a positive semidefinite matrix
with simple least eigenvalue and distinct eigenvalues, which can provide an
algebraic characterization to their problem with respect to the (normalized)
Laplacian matrix.Comment: 11 page
On Products and Line Graphs of Signed Graphs, their Eigenvalues and Energy
In this article we examine the adjacency and Laplacian matrices and their
eigenvalues and energies of the general product (non-complete extended -sum,
or NEPS) of signed graphs. We express the adjacency matrix of the product in
terms of the Kronecker matrix product and the eigenvalues and energy of the
product in terms of those of the factor signed graphs. For the Cartesian
product we characterize balance and compute expressions for the Laplacian
eigenvalues and Laplacian energy. We give exact results for those signed
planar, cylindrical and toroidal grids which are Cartesian products of signed
paths and cycles.
We also treat the eigenvalues and energy of the line graphs of signed graphs,
and the Laplacian eigenvalues and Laplacian energy in the regular case, with
application to the line graphs of signed grids that are Cartesian products and
to the line graphs of all-positive and all-negative complete graphs.Comment: 30 page
Bounds on the negative eigenvalues of Laplacians on finite metric graphs
For a self--adjoint Laplace operator on a finite, not necessarily compact,
metric graph lower and upper bounds on each of the negative eigenvalues are
derived. For compact finite metric graphs Poincar\'{e} type inequalities are
given.Comment: 17 page
- …