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

TL;DR

This paper determines the maximum inverse sum indeg (ISI) bond incident degree index for trees and unicyclic graphs with fixed diameters, identifying extremal structures using graph transformation methods.

Contribution

It establishes the extremal graphs that maximize the ISI index among trees and unicyclic graphs with given diameters, expanding understanding of graph invariants.

Findings

Maximum ISI index for trees is attained by a specific tree structure.

Extremal unicyclic graphs vary with diameter, identified as specific known graphs.

Provides characterizations for extremal graphs based on 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 \(d \geq 4\).