Trees with minimal Laplacian coefficients

Let G be a simple undirected graph with the characteristic polynomial of its Laplacian matrix L.G/, P.G; / D Pn kD0.􀀀1/kck n􀀀k. It is well known that for trees the Laplacian coefficient cn􀀀2 is equal to the Wiener index of G, while cn􀀀3 is equal to the modified hyper-Wiener index of the graph. In this paper, we characterize n-vertex trees with given matching number m which simultaneously minimize all Laplacian coefficients. The extremal tree A.n;m/ is a spur, obtained from the star graph Sn􀀀mC1 with n􀀀mC1 vertices by attaching a pendant edge to each of certain m 􀀀 1 non-central vertices of Sn􀀀mC1. In particular, A.n;m/ minimizes the Wiener index, the modified hyper-Wiener index and the recently introduced Incidence energy of trees, defined as IE.G/ D Pn kD0 p k, where k are the eigenvalues of signless Laplacian matrix Q.G/ D D.G/CA.G/. We introduced a general transformation which decreases all Laplacian coefficients simultaneously. In conclusion, we illustrate on examples of Wiener index and Incidence energy that the opposite problem of simultaneously maximizing all Laplacian coefficients has no solution.