On the Laplacian coefficients of bicyclic graphs

Let GG be a graph of order nn and let P(G,x)=∑k=0n(−1)kckxn−k be the characteristic polynomial of its Laplacian matrix. Generalizing the approach in [D. Stevanović, A. Ilić, On the Laplacian coefficients of unicyclic graphs, Linear Algebra and its Applications 430 (2009) 2290–2300.] on graph transformations, we show that among all bicyclic graphs of order nn, the kkth coefficient ckck is smallest when the graph is BnBn (obtained from C4C4 by adding one edge connecting two non-adjacent vertices and adding n−4n−4 pendent vertices attached to the vertex of degree 3).