The largest Laplacian and signless Laplacian H-eigenvalues of a uniform hypergraph

In this paper, we show that the largest Laplacian H-eigenvalue of a k-uniform nontrivial hypergraph is strictly larger than the maximum degree when k is even. A tight lower bound for this eigenvalue is given. For a connected even-uniform hypergraph, this lower bound is achieved if and only if it is a hyperstar. However, when k is odd, in certain cases the largest Laplacian H-eigenvalue is equal to the maximum degree, which is a tight lower bound. On the other hand, tight upper and lower bounds for the largest signless Laplacian H-eigenvalue of a k-uniform connected hypergraph are given. For connected k-uniform hypergraphs of fixed number of vertices (respectively fixed maximum degree), the upper (respectively lower) bound of their largest signless Laplacian H-eigenvalues is achieved exactly for the complete hypergraph (respectively the hyperstar). The largest Laplacian H-eigenvalue is always less than or equal to the largest signless Laplacian H-eigenvalue. When the hypergraph is connected, the equality holds here if and only if k is even and the hypergraph is odd-bipartite.