On The Principal Eigenvector Of Graph And Its Application

In this paper, we mainly study the sharp bounds of components in the principle eigenvector with maximal and minimal modulus corresponding to Laplacian spectral radius of simple undirected graphs. By analyzing the structure properties of graphs, we characterize the extremal graphs with the sharp upper bounds of entries in the principle eigenvector with maximal modulus.Furthermore, we discuss the extremal graphs on n vertices with fixed numbers of blocks which have the largest spectral radius by graft transformation. We also give ordering of spectral radius of other special classes of graphs. The main results are as follows:1 We discuss the sharp bounds of entries in the principle eigenvector with maximal modulus corresponding to Laplacian matrix of graphs, then give the extremal graphs with the sharp bounds. Particularly, we give all the extremal graphs when G is bipartite graph.2 We show that the maximal spectral radius of graphs with fixed numbers of blocks is obtained uniquely obtained from K_n-k+1 by joining k - 1 pendant vertices to a same vertex of K_n-k+1. We also give the extremal graphs of graphs on fixed numbers of blocks with maximal Laplacian spectral radius and graphs on fixed number of blocks and largest cut degrees with maximal spectral radius.