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

TL;DR

This paper characterizes trees and connected graphs with given diameter or radius that minimize all Laplacian coefficients, linking these coefficients to the Wiener index and extending previous work on graph optimization.

Contribution

It generalizes existing results by characterizing graphs with fixed diameter or radius that minimize Laplacian coefficients, connecting these to Wiener index minimization.

Findings

Trees with fixed diameter that minimize Laplacian coefficients are characterized.

Connected graphs with fixed radius minimizing Laplacian coefficients are identified.

Maximizing all Laplacian coefficients simultaneously has no solution.

Abstract

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.