Xiying Yuan

On the principal eigenvectors of uniform hypergraphs

Abstract Let A ( H ) be the adjacency tensor of r-uniform hypergraph H. If H is connected, the unique positive eigenvector x = ( x 1 , x 2 , ⋯ , x n ) 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 x max and x min , respectively. In this paper, we investigate the bounds of x max and x min in the principal eigenvector of H. Meanwhile, we also obtain some bounds of the ratio x i / x j for i, j ∈ [ n ] as well as the principal ratio γ ( H ) = x max / x min 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 ′ .

Maxima of the Q-index: graphs without long paths

This paper gives tight upper bound on the largest eigenvalue q(G) of the signless Laplacian of graphs with no paths of given order. The main ingredient of our proof is a stability result of its own interest, about graphs with large minimum degree and with no long paths. This result extends previous work of Ali and Staton.

Some properties and applications of odd-colorable r-hypergraphs

Abstract Let r ≥ 2 and r be even. An r -hypergraph G on n vertices is called odd-colorable if there exists a map φ : [ n ] → [ r ] such that for any edge { j 1 , j 2 , … , j r } of G , we have φ ( j 1 ) + φ ( j 2 ) + ⋅ ⋯ ⋅ + φ ( j r ) ≡ r ∕ 2 ( mod r ) . In this paper, we first determine that, if r = 2 q ( 2 t + 1 ) and n ≥ 2 q ( 2 q − 1 ) r , then the maximum chromatic number in the class of the odd-colorable r -hypergraphs on n vertices is 2 q , which answers a question raised by V. Nikiforov recently in Nikiforov (2017). We also study some applications of the spectral symmetry of the odd-colorable r -hypergraphs given in the same paper by V. Nikiforov. We show that the Laplacian spectrum S p e c ( L ( G ) ) and the signless Laplacian spectrum S p e c ( Q ( G ) ) of an r -hypergraph G are equal if and only if r is even and G is odd-colorable. As an application of this result, we give an affirmative answer for the remaining unsolved case r ⁄ ≡ 0 ( m o d 4 ) of a question raised in Shao et al. (2015) about whether S p e c ( L ( G ) ) = S p e c ( Q ( G ) ) implies that L ( G ) and Q ( G ) have the same H-spectrum.

Spectral radii of two kinds of uniform hypergraphs

Abstract Let A ( H ) be the adjacency tensor (hypermatrix) of uniform hypergraph H. The maximum modulus of the eigenvalues of A ( H ) is called the spectral radius of H, denoted by ρ(H). In this paper, a conjecture concerning the spectral radii of linear bicyclic uniform hypergraphs is solved, with these results the hypergraph with the largest spectral radius is completely determined among the linear bicyclic uniform hypergraphs. For a t-uniform hypergraph G its generalized power r-uniform hypergraph Gr, s is defined in this paper. An exact relation between ρ(G) and ρ(Gr, s) is proved, more precisely ρ ( G r , s ) = ( ρ ( G ) ) t s r .