Extremal Halin graphs with respect to the signless Laplacian spectra

A Halin graph G is a plane graph constructed as follows: Let T be a tree on at least 4 vertices. All vertices of T are either of degree 1, called leaves, or of degree at least 3. Let C be a cycle connecting the leaves of T in such a way that C forms the boundary of the unbounded face. Denote the set of all n-vertex Halin graphs by Gn. In this article, sharp upper and lower bounds on the signless Laplacian indices of graphs among Gn are determined and the extremal graphs are identified, respectively. As well graphs in Gn having the second and third largest signless Laplacian indices are determined, respectively.