On the signless Laplacian coefficients of unicyclic graphs

Let G be a graph of order n and let QG(x)=∑i=0n(−1)ipi(G)xn−i be the characteristic polynomial of the signless Laplacian of G. Let Eg,n (respectively, Cg(Sn−g+1)) denote the unicyclic graph of order n obtained by a coalescence of a vertex in the cycle Cg with an end vertex (respectively, the center) of the path Pn−g+1 (respectively, the star Sn−g+1). It is proved that for k=2,…,n−1, as G varies over all unicyclic graphs of order n, depending on k and n, the maximum value of pk(G) is attained at G=Cn or E3,n, and the minimum value is attained uniquely at G=C4(Sn−3) or C3(Sn−2). Except for the resolution of a conjecture on cubic polynomials, the uniqueness issue for the maximization problem is also settled.