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 $\text{BID}(G)={\sum }_{u_1{u}_2\in E(G)}f(d(u_1),d(u_2))$, 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 $d\ge 3$ and $n\ge d+3$, the maximum ISI index is attained by the tree $T_{n,d}^{\ast }$. For unicyclic graphs, we characterize the extremal graphs for diameters $d=2$, $d=3$, and $d\ge 4$. Specifically, the maximum ISI index is achieved by $S_n^{+}$ for $d=2$, by $C_n^{\ast }$ for $d=3$, and by ${\mathcal{U}}_{n,d}$ for $d\ge 4$.