  Let $G$ be a simple undirected $n$-vertex graph with the characteristic
polynomial of its Laplacian matrix $L(G)$, $\det (\lambda I - L (G))=\sum_{k =
0}^n (-1)^k c_k \lambda^{n - k}$. It is well known that for trees the Laplacian
coefficient $c_{n-2}$ is equal to the Wiener index of $G$. Using a result of
Zhou and Gutman on the relation between the Laplacian coefficients and the
matching numbers in subdivided bipartite graphs, we characterize first the
trees with given diameter and then the connected graphs with given radius which
simultaneously minimize all Laplacian coefficients. This approach generalizes
recent results of Liu and Pan [MATCH Commun. Math. Comput. Chem. 60 (2008),
85--94] and Wang and Guo [MATCH Commun. Math. Comput. Chem. 60 (2008),
609--622] who characterized $n$-vertex trees with fixed diameter $d$ which
minimize the Wiener index. In conclusion, we illustrate on examples with Wiener
and modified hyper-Wiener index that the opposite problem of simultaneously
maximizing all Laplacian coefficients has no solution.
