The signless Laplacian coefficients and incidence energy of bicyclic graphs

Let Q(G;x)=det(xI−Q(G))=∑i=1n(−1)iφixn−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)Q(G;x)) change after some graph transformations. These results are used to prove that the set (Bn,⪯)(Bn,⪯) 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.