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Let $$\mathbb{S}$$ S ( m, d, k ) be the set of k -uniform supertrees with m edges and diameter d , and S 1 ( m, d, k ) be the k -uniform supertree obtained from a loose path u 1 , e 1 , u 2 , e 2 , …, 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 S 1 ( m, d, k ) with the largest signless Laplacian spectral radius in $$\mathbb{S}$$ S ( m, d, k ) for 3 ⩽ d ⩽ m − 1. We also determine the supertree with the second largest signless Laplacian spectral radius in $$\mathbb{S}$$ 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).

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