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

Abstract

Let $S_1(m,d,k)$ be the $k$-uniform supertree obtained from a loose path $P:v_1,e_1,v_2,\dots ,v_d,e_d,v_{d+1}$ with length $d$ by attaching $m-d$ edges at vertex $v_{\lfloor \frac{d}{2}\rfloor +1}.$ Let $\mathbb{S}(m,d,k)$ be the set of $k$-uniform supertrees with $m$ edges and diameter $d$ and $q(G)$ be the signless Laplacian spectral radius of a $k$-uniform hypergraph $G$. In this paper, we mainly determine $S_1(m,d,k)$ with the largest signless Laplacian spectral radius among all supertrees in $\mathbb{S}(m,d,k)$ for $3\le d\le m-1$. Furthermore, we determine the unique uniform supertree with the maximum signless Laplacian spectral radius among all the uniform supertrees with $n$ vertices and pendent edges (vertices).

12 pages, 2 figures