On the Wiener index and Laplacian coefficients of graphs with given\n diameter or radius

Let $G$ be a simple undirected $n$-vertex graph with the characteristic\npolynomial of its Laplacian matrix $L(G)$, $\\det (\\lambda I - L (G))=\\sum_{k =\n0}^n (-1)^k c_k \\lambda^{n - k}$. It is well known that for trees the Laplacian\ncoefficient $c_{n-2}$ is equal to the Wiener index of $G$. Using a result of\nZhou and Gutman on the relation between the Laplacian coefficients and the\nmatching numbers in subdivided bipartite graphs, we characterize first the\ntrees with given diameter and then the connected graphs with given radius which\nsimultaneously minimize all Laplacian coefficients. This approach generalizes\nrecent results of Liu and Pan [MATCH Commun. Math. Comput. Chem. 60 (2008),\n85--94] and Wang and Guo [MATCH Commun. Math. Comput. Chem. 60 (2008),\n609--622] who characterized $n$-vertex trees with fixed diameter $d$ which\nminimize the Wiener index. In conclusion, we illustrate on examples with Wiener\nand modified hyper-Wiener index that the opposite problem of simultaneously\nmaximizing all Laplacian coefficients has no solution.\n