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).