  In this paper we show how the $\alpha$-spectral radius changes under the edge
grafting operations on connected $k$-uniform hypergraphs. We characterize the
extremal hypertree for $\alpha$-spectral radius among $k$-uniform
non-caterpillar hypergraphs with given order, size and diameter. we also
characerize the second largest $\alpha$-spectral radius among all $k$-uniform
supertrees on $n$ vertices by two methods.
