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 G n . In this article, sharp upper and lower bounds on the signless Laplacian indices of graphs among G n are determined and the extremal graphs are identified, respectively. As well graphs in G n having the second and third largest signless Laplacian indices are determined, respectively.