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, it happens that 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 a
connected k-uniform hypergraph, the upper (respectively lower) bound of the
largest signless Laplacian H-eigenvalue is achieved if and only if it is a
complete hypergraph (respectively a 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.Comment: 26 pages, 3 figure