Maximum Inverse Sum Indeg Index of Trees and Unicyclic Graphs with Fixed Diameter
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914386255282176 |
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| author | Hosamani, Sunilkumar M. |
| author_facet | Hosamani, Sunilkumar M. |
| contents | 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 \(d \geq 4\). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_10603 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Maximum Inverse Sum Indeg Index of Trees and Unicyclic Graphs with Fixed Diameter Hosamani, Sunilkumar M. Combinatorics 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 \(d \geq 4\). |
| title | Maximum Inverse Sum Indeg Index of Trees and Unicyclic Graphs with Fixed Diameter |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.10603 |