Maximum Inverse Sum Indeg Index of Trees and Unicyclic Graphs with Fixed Diameter

Abstract

The bond incident degree (BID) index of a graph \(G\) is defined as \(\BID(G) = \sum_{u_1u_2\in E(G)} f(d(u_1), d(u_2))\), where \(f(x,y)=f(y,x)\) is a real-valued function. In this paper, using graph transformation methods, we establish the maximum bond incident degree indices of trees and unicyclic graphs with a fixed diameter for the inverse sum indeg (ISI) index. The ISI index corresponds to the function \(f(x,y) = \frac{xy}{x+y}\). We prove that for trees \(T \in \mathbb{T}_{n,d}\) with \(d \geq 3\) and \(n \geq d+3\), the maximum ISI index is attained by the tree \(T_{n,d}^*\). For unicyclic graphs, we characterize the extremal graphs for diameters \(d=2\), \(d=3\), and \(d \geq 4\). Specifically, the maximum ISI index is achieved by \(S_n^+\) for \(d=2\), by \(C_n^*\) for \(d=3\), and by \(\mathcal{U}_{n,d}\) for \(d \geq 4\).