树的第二大特征值的界

Bounds on the Second Largest Eigenvalues of Trees

在文献 Ji-Ming Guo, Shang-Wang Tan. A conjecture on the second largest eigenvalue of a tree with perfect matchings. Linear Algebra and its Applications, 2002, 347（1-3）： 9-15和Ji-Ming Guo, Shang-Wang Tan. A note on the second largest eigenvalue of a tree with perfect matchings. Linear Algebra and Its Applications, 2004, 380：125-134中，Guo和Tan给出了有2k个顶点且有完美匹配的树的第二大特征值的上界，这个上界与顶点数有关，并且刻画了第二大特征值达到该上界的树．本文给出了有2k个顶点的树的第二大特征值的上界，这个上界与顶点数和最大匹配的基数有关，并且刻画了第二大特征值达到该上界的树．

Guo and Tan in [Ji-Ming Guo, Shang-Wang Tan, A conjecture on the second largest eigenvalue of a tree with perfect matchings, Linear Algebra and its Applications 347 （1-3） （2002） 9-15] and [Ji-Ming Guo, Shang-Wang Tan, A note on the second largest eigenvalue of a tree with perfect matchings, Linear Algebra and its Applications 880 （2004） 125-134] presented the upper bounds for the second largest eigenvalue of a tree on 2k vertices with perfect matchings in terms of the number of vertices and characterized the trees whose second largest eigenvalues attain the upper bounds. In this paper, we present the upper bounds for the second largest eigenvalue of a tree on 2k vertices in terms of the number of vertices and the size of maximum matchings and characterize the trees whose second largest eigenvalues attain the upper bounds.