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

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 $d \geq 4$.