The Laplacian energy and Laplacian eigenvalues of the K-trees

Abstract

A k-tree is either a complete graph on k vertices or a graph obtained from a smaller k-tree by adjoining a new vertex together with k edges connecting it to a k-clique. Denote the set of all n-vertex k-trees by T k n. In this paper, we impose some restrictions on the spectrum of a k-tree with the k number has largest Laplacian energy and smallest Laplacian energy among all the graphs satisfying those conditions. The corresponding extremal graphs are characterized respectively as well.