The biharmonic index of connected graphs

Let $ G $ be a simple connected graph with the vertex set $ V(G) $ and $ d_{B}(u, v) $ be the biharmonic distance between two vertices $ u $ and $ v $ in $ G $. The biharmonic index $ BH(G) $ of $ G $ is defined as \begin{document}$ BH(G) = \frac{1}{2}\sum\limits_{u\in V(G)}\sum\limits_{v\in V(G)}d_{B}^2(u, v) = n\sum\limits_{i = 2}^{n}\frac{1}{\lambda_i^2(G)}, $\end{document} where $ \lambda_i(G) $ is the $ i $-th eigenvalue of the Laplacian matrix of $ G $ with $ n $ vertices. In this paper, we provide the mathematical relationships between the biharmonic index and some classic topological indices: the first Zagreb index, the forgotten topological index and the Kirchhoff index. In addition, the extremal value on the biharmonic index for all graphs with diameter two, trees and firefly graphs are given, respectively. Finally, some graph operations on the biharmonic index are presented.