A subdivision theorem for the largest signless Laplacian eigenvalue of a graph

Let n >= 2 be an integer and G be a graph with a vertex v such that d(v) >= 2n. Partition the vertices of G adjacent to v into n sets W-1, ... , W-n, where each W-i contains at least 2 vertices. Let the vertices of the complete graph K-n be labeled v(1), ... , v(n), and G(Kn), be the graph isomorphic to K-n boolean OR (G \ v) along with edges joining the vertices of W-i with v(i) (for i = 1, ... , n). Denote by q(G) the largest signless Laplacian eigenvalue of G. In this paper we prove that q(G(Kn)) < q(G). As an application, we give a corollary which is similar to the result of Simic about splitting vertices.