On some properties of the $\alpha$-spectral radius of the $k$-uniform hypergraph

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.