On the Laplacian coefficients of tricyclic graphs

Let Φ ( G , λ ) = d e t ( λ I n − L ( G ) ) = ∑ k = 0 n ( − 1 ) k c k λ n − k be the characteristic polynomial of the Laplacian matrix of a graph G of order n . In this paper, we show that among all connected tricyclic graphs of order n , the k th coefficient c k is smallest when the graph is B n , 7 ( 1 ) 3 , 3 , 3 (obtained from the complete graph K 4 by adding n − 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.