On the α-spectra of Uniform Hypergraphs and Its Associated Graphs

For 0 ≤α < 1 and a k-uniform hypergraph H, the tensor Aα(H) associated with H is defined as Aα(H) = αD(H) +(1-α)A(H), where D(H) and A(H) are the diagonal tensor of degrees and the adjacency tensor of H, respectively. The α-spectra of H is the set of all eigenvalues of Aα(H) and the α-spectral radius ρα(H) is the largest modulus of the elements in the spectrum of Aα(H). In this paper we define the line graph L(H) of a uniform hypergraph H and prove that ρα(H) ≤■ρα(L(H)) + 1 + α(Δ-1-δ*/k), where Δ and δ*are the maximum degree of H and the minimum degree of L(H), respectively. We also generalize some results on α-spectra of Gk,s, which is obtained from G by blowing up each vertex into an s-set and each edge into a k-set where 1 ≤ s ≤ k/2.