The proof of a conjecture on largest Laplacian and signless Laplacian H-eigenvalues of uniform hypergraphs

Let A(G),L(G)A(G),L(G) and Q(G)Q(G) be the adjacency tensor, Laplacian tensor and signless Laplacian tensor of uniform hypergraph G  , respectively. Denote by λ(T)λ(T) the largest H-eigenvalue of tensor TT. Let H   be a uniform hypergraph, and H′H′ be obtained from H   by inserting a new vertex with degree one in each edge. We prove that λ(Q(H′))≤λ(Q(H))λ(Q(H′))≤λ(Q(H)). Denote by GkGk the kth power hypergraph of an ordinary graph G   with maximum degree Δ≥2Δ≥2. We prove that {λ(Q(Gk))}{λ(Q(Gk))} is a strictly decreasing sequence, which implies Conjecture 4.1 of Hu, Qi and Shao in [4]. We also prove that λ(Q(Gk))λ(Q(Gk)) converges to Δ when k goes to infinity. The definition of k  th power hypergraph GkGk has been generalized as Gk,sGk,s. We also prove some eigenvalues properties about A(Gk,s)A(Gk,s), which generalize some known results. Some related results about L(G)L(G) are also mentioned.