On the principal eigenvectors of uniform hypergraphs

Let A(H) be the adjacency tensor of r-uniform hypergraph H. If H is connected, the unique positive eigenvector x=(x1,x2,⋯,xn)T with ‖x‖r=1 corresponding to spectral radius ρ(H) is called the principal eigenvector of H. The maximum and minimum entries of x are denoted by xmax and xmin, respectively. In this paper, we investigate the bounds of xmax and xmin in the principal eigenvector of H. Meanwhile, we also obtain some bounds of the ratio xi/xj for i, j∈[n] as well as the principal ratio γ(H)=xmax/xmin of H. As an application of these results we finally give an estimate of the gap of spectral radii between H and its proper sub-hypergraph H′.