Some Studies On Spectral Of Hypergraphs

Studying the relationship between the eigenvalue of graph and the structure of graph has been the mainstream problems in the theory of graph spectral. Hypergraph is a new development of graph. As the theory of the eigenvalue of tensor developed recently, the theory of hypergraph spectra becomes more and more interesting.In 2012, J. Cooper and A. Dutle defined the adjacency tensor of uniform hypergraph. In 2014, L. Qi defined the Laplacian tensor and signless Laplacian tensor of uniform hypergraph.The research of adjacency tensor, Laplacian tensor and signless Laplacian tensor arouse many scholars’ attentions . In this paper, we show that a connected uniform hypergraph G is odd-bipartite if and only if G has the same Laplacian and signless Laplacian Z -eigenvalue.We obtain some bounds for the largest (signless) Laplacian Z -eigenvalue of a hypergraph asfollows: Δ≤λ(LG)≤λ(QG)≤2Δ and,where Δ isbiggest degree of G, LG and QG be the Laplacian tensor and signless Laplacian tensor of G . For a k-uniform hyperstar S(d,k) with d edges. When 2d ≥ k ≥ 3, we show that its largest (signless) Laplacian Z -eigenvalue is d.In 2015, H. Li, J. Shao and L. Qi defined the incidence Q-tensor of uniform hypergraph and studied some extremal problems of three kinds of spectal radii which consists of adjacency tensor, signless Laplacian and incidence Q-tensor. In this paper, we provide some properties of H-eigenvalue of Q-tensor of cored hypergraph and power hpergraph. We further show all H-eigenvalues of Q-tensor of even uniform hyprestar with d edges are(d1/k-1+k-1)k-1,0,(k-1)k-1 and d.