The biharmonic index of connected graphs

TL;DR

This paper introduces the biharmonic index of connected graphs, explores its mathematical relationships with classic topological indices, and determines extremal values for specific graph classes.

Contribution

It defines the biharmonic index, relates it to known indices, and analyzes its extremal properties for trees and firefly graphs.

Findings

Established relationships between biharmonic index and topological indices

Identified extremal biharmonic index values for trees and firefly graphs

Presented graph operations affecting the biharmonic index

Abstract

Let $G$ be a simple connected graph with the vertex set $V(G)$ and $d_{B}^2(u,v)$ be the biharmonic distance between two vertices $u$ and $v$ in $G$. The biharmonic index $BH(G)$ of $G$ is defined as $$BH(G)=\frac{1}{2}\sum\limits_{u\in V(G)}\sum\limits_{v\in V(G)}d_{B}^2(u,v)=n\sum\limits_{i=2}^{n}\frac{1}{\lambda_i^2(G)},$$ where $\lambda_i(G)$ is the $i$-th smallest 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 trees and firefly graphs of fixed order are given. Finally, some graph operations on the biharmonic index are presented.