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

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