On a Problem of Harary and Schwenk on Graphs with Distinct Eigenvalues
Combinatorics
2014-05-26 v1
Abstract
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.
Keywords
Cite
@article{arxiv.1405.6000,
title = {On a Problem of Harary and Schwenk on Graphs with Distinct Eigenvalues},
author = {Xueliang Li and Jianfeng Wang and Qiongxiang Huang},
journal= {arXiv preprint arXiv:1405.6000},
year = {2014}
}
Comments
11 pages