关于具有不同特征值的图的Harary和Schwenk问题
On a Problem of Harary and Schwenk on Graphs with Distinct Eigenvalues
摘要：
Harary和Schwenk在40年前提出了这个问题：哪些图有不同的邻接特征值？本文给出了谱半径简单且特征值不同的埃尔米特矩阵的一个充要条件。作为其应用，我们给出了Harary-Schwenk问题的一个代数刻画。作为其问题的推广，我们还得到了正半定矩阵具有简单最小特征值和不同特征值的一个充要条件，从而给出了关于（正规化）Laplacian矩阵的问题的一个代数刻画.
英文摘要：
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.
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