On the signless Laplacian spectra of k-trees

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 Tnk. In this paper, the graphs among Tnk having the first (resp. the second and the third) largest signless Laplacian index q1(G)q1(G) are characterized and the sharp upper bound on the signless Laplacian index of graphs in Tnk is determined. Furthermore, sharp upper bound on q1(G)q1(G) (resp. q1(G)+k,q1(G)−k,q1(G)⋅k,q1(G)/kq1(G)+k,q1(G)−k,q1(G)⋅k,q1(G)/k) of graph G   in ⋃k=1n−1Tnk are determined. The corresponding extremal graphs are characterized respectively as well.