Tree with Minimal Laplacian Spectral Radius and Diameter \(n-4\)

Abstract

Let \(\mathcal{T}_{n,n-4}\) be the set of trees on \(n\) vertices with diameter \(n-4\). In this paper, we determine the unique tree which has the minimal Laplacian spectral radius among all trees in \(\mathcal{T}_{n,n-4}\).
This work is related to that of Yuan [The minimal spectral radius of graphs of order n with diameter \(n – 4\), Linear Algebra Appl. \(428(2008)2840-2851]\), which determined the graph with minimal spectral radius among all the graphs of order \(n\) with diameter \(n-4\). We can observe that the extremal tree on the Laplacian spectral radius is different from that on the spectral radius.