Largest signless Laplacian spectral radius of uniform supertrees with diameter and pendent edges (vertices)

Let S(m, d, k) be the set of k-uniform supertrees with m edges and diameter d, and S₁ (m, d, k) be the k-uniform supertree obtained from a loose path u₁, e₁, u₂, e₂, …, u_d, e_d, u_d+1 with length d by attaching m–d edges at vertex u_⌊d/2⌋+1. In this paper, we mainly determine S₁ (m, d, k) with the largest signless Laplacian spectral radius in S(m, d, k) for 3 ⩽ d ⩽ m − 1. We also determine the supertree with the second largest signless Laplacian spectral radius in S(m, 3, k). Furthermore, we determine the unique k-uniform supertree with the largest signless Laplacian spectral radius among all k-uniform supertrees with n vertices and pendent edges (vertices).