On the signless Laplacian coefficients of unicyclic graphs

Let G be a graph of order n and let Q(G)(x)= Sigma(n)(i=0)(-1)(i)p(i)(G)x(n-i) be the characteristic polynomial of the signless Laplacian of G. Let E-g,E-n (respectively, C-g(Sn-g+1)) denote the unicyclic graph of order n obtained by a coalescence of a vertex in the cycle C-g 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 p(k)(G) is attained at G = C-n or E-3,E-n, and the minimum value is attained uniquely at G = C-4(Sn-3) or C-3 (Sn-2). Except for the resolution of a conjecture on cubic polynomials, the uniqueness issue for the maximization problem is also settled. (C) 2013 Elsevier Inc. All rights reserved.