On the Laplacian coefficients of unicyclic graphs

Abstract

Let G be a graph of order n and let P (G, λ) = ∑k = 0n (- 1)k ck λn - k be the characteristic polynomial of its Laplacian matrix. Generalizing an approach of Mohar on graph transformations, we show that among all connected unicyclic graphs of order n, the kth coefficient ck is largest when the graph is a cycle Cn and smallest when the graph is the a Sn with an additional edge between two of its pendent vertices. A relation to the recently established Laplacian-like energy of a graph is discussed.