On the Laplacian coefficients of tricyclic graphs

Let Φ(G,λ)=det(λIn−L(G))=∑k=0n(−1)kckλn−k be the characteristic polynomial of the Laplacian matrix of a graph GG of order nn. In this paper, we show that among all connected tricyclic graphs of order nn, the kkth coefficient ckck is smallest when the graph is Bn,7(1)3,3,3 (obtained from the complete graph K4K4 by adding n−4n−4 pendent vertices attached to the vertex of degree 3). And for some lemmas in [C. X. He, H. Y. Shan, On the Laplacian coefficients of bicyclic graphs, Discrete Math. 310 (2010) 3404–3412], we present a new method to prove them.