On the Laplacian coefficients of bicyclic graphs

Abstract

Let G be a graph of order n and let P ( G , x ) = ∑ k = 0 n ( − 1 ) k c k x n − 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 n , the k th coefficient c k is smallest when the graph is B n (obtained from C 4 by adding one edge connecting two non-adjacent vertices and adding n − 4 pendent vertices attached to the vertex of degree 3).