The biharmonic index of connected graphs

<abstract><p>Let $ G be a simple connected graph with the vertex set V(G) and d_{B}(u, v) biharmonic distance between two vertices u v in $. The index BH(G) of is defined as</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ = \frac{1}{2}\sum\limits_{u\in V(G)}\sum\limits_{v\in V(G)}d_{B}^2(u, n\sum\limits_{i 2}^{n}\frac{1}{\lambda_i^2(G)}, $\end{document} </tex-math></disp-formula></p> <p>where \lambda_i(G) i $-th eigenvalue Laplacian matrix n vertices. In this paper, we provide mathematical relationships some classic topological indices: first Zagreb index, forgotten Kirchhoff index. addition, extremal value on for all graphs diameter two, trees firefly are given, respectively. Finally, operations presented.</p></abstract>