Bounds Of Eigenvalues Of P-laplacian Of The Graph

The contribution of this thesis is to improve the bound of p-Laplacian and to generalize a work of Xiaodong Zhang on Laplacian to p-Laplacian.And to find the relationship between the characteristic components corresponding to the eigenvalues of a class of trees T_n,kwhich is not equal to 1.The work of this paper mainly includes as follow:1.We improve the result of S.Amghibech about the maximum p-Laplacian upper bound of the graph to,??2^p-1max{(₂^du+dv),[u,v]?E},where?is the largest p-Laplacian eigenvalue of the graph G.And we prove that the result of this paper are better than that given by S.Amghibech.2.Recently,Xiaodong Zhang has given the following upper bound for the Lapla-cian eigenvalue:??G??max{d?u??d?u?+m?u??+d?v??d?v?+m?v??,[u,v]?E}.We generalize it to p-Laplacian as??max{2pq-1[d?u?q?d?u?q+u^xdd??xu??q?+d?v?q?d?v?q+v^ydd??yv??q?]1q,[u,v]?E}.3.Through the analysis of the characteristics of the path we prove that the characteristic components corresponding to the p-Laplacian eigenvalues of the sym-metric position of the path with odd number of vertices are equal.We prove that the characteristic components corresponding to the p-Laplacian eigenvalues of vertices at symmetric positions of a class of tree T_n,kare equal.