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

Sunilkumar M. Hosamani

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 \(d \geq 4\).

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

Abstract

The bond incident degree (BID) index of a graph 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 with and , the maximum ISI index is attained by the tree . For unicyclic graphs, we characterize the extremal graphs for diameters , , and . Specifically, the maximum ISI index is achieved by for , by for , and by for .
Paper Structure (8 sections, 11 theorems, 23 equations, 7 figures, 2 tables)

This paper contains 8 sections, 11 theorems, 23 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Maximal ISI Index of Trees with Fixed Diameter
  3. Maximal ISI Index of Unicyclic Graphs with Fixed Diameter
  4. Diameter $d = 2$
  5. Diameter $d = 3$
  6. Diameter $d \geq 4$
  7. Application to the Inverse Sum Indeg Index
  8. Conclusion

Key Result

Lemma 2.5

Denote by $P_l = v_1v_2\ldots v_l$ an induced subpath of graph $G$ such that $d_G(v_1) \geq 2$ and $d_G(v_l) \geq 2$. Let $G' = G - \sum_{u \in N_G(v_l)\setminus v_{l-1}} v_l u + \sum_{u \in N_G(v_l)\setminus v_{l-1}} v_1 u$. The operation of constructing $G'$ based on $G$ is called the path lifting then $\operatorname{BID}(G) < \operatorname{BID}(G')$.

Figures (7)

  • Figure 1: Tree structure with pendant vertices attached to interior vertices of the diameter path
  • Figure 2: Path lifting transformation: moving pendant vertices from $v_5$ to $v_1$
  • Figure 3: The extremal tree $T_{n,d}^i$ with all pendant vertices attached to a single interior vertex $u_i$
  • Figure 4: Tree transformation when high-degree vertices are adjacent
  • Figure 5: Special unicyclic graphs: $S_n^+$, $C_n^*$, and $B_n^*$ with highlighted cycle vertices
  • ...and 2 more figures

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5: Su su2025extremal
  • Lemma 2.6
  • proof
  • Theorem 2.7
  • proof
  • Definition 3.1
  • ...and 17 more