The Quasi-tree Graph with Maximum Laplacian Spread

Abstract

The Laplacian spread of a graph G is defined to be the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of G. Let $${\mathcal {Q}}_t(n, d)=\{G\, |\, G-v_0\ \hbox {is a tree on }n-1\hbox { vertices and } d_G(v_0)=d\}.$$ Recently, Y. Xu and J. Meng characterized the unique graph with maximum Laplacian spread among all graphs in the set $${\mathcal {Q}}_t(n, d)$$ with $$1\le d\le (n-4)/2$$ . In this paper, we extend their result by determining the unique graph with maximum Laplacian spread among all graphs in the set $${\mathcal {Q}}_t(n, d)$$ with $$1\le d\le n-5$$ .