The signless Laplacian coefficients and incidence energy of bicyclic graphs

Let Q (G; x) = det(xI - Q (G)) = E-i=1(n) (-1)i phi(i)x(n-i) be the characteristic polynomial of the signless Laplacian.matrix of a graph G of order n. This paper investigates how the signless Laplacian coefficients (i.e., coefficients of Q(G; x)) change after some graph transformations. These results are used to prove that the set (B-n, <=) of all bicyclic graphs of order n has exactly two minimal elements with respect to the partial ordering of their coefficients. Furthermore, we present a sharp lower bound for the incidence energy of bicyclic graphs of order n and characterize all extremal graphs. (C) 2013 Elsevier Inc. All rights reserved.