Improved Lower Bounds on the General Reduced Second Zagreb Index of Trees and Unicyclic Graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914614806052864 |
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| author | Dehgardi, Nasrin Klavžar, Sandi Azari, Mahdieh |
| author_facet | Dehgardi, Nasrin Klavžar, Sandi Azari, Mahdieh |
| contents | For a simple graph $Γ$ and a real number $λ$, the general reduced second Zagreb index is defined by the formula $$GRM_λ(Γ)=\sum_{ab\in E(Γ)}[(°_Γ(a)+λ)(°_Γ(b)+λ)]\,.$$ A sharp lower bound for $GRM_λ$ over all trees of given order and maximum degree under the condition that $λ\ge -\frac{1}{2}$ is established. A parallel result is proved for unicyclic graphs under the condition $λ\ge -\frac{1}{2}$. The corresponding minimal trees and unicyclic graphs are identified. These findings improve upon the lower bounds previously established by Buyantogtokh, Horoldagva, and Das concerning $GRM_λ$ of trees and unicyclic graphs of given order. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_30402 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Improved Lower Bounds on the General Reduced Second Zagreb Index of Trees and Unicyclic Graphs Dehgardi, Nasrin Klavžar, Sandi Azari, Mahdieh Combinatorics For a simple graph $Γ$ and a real number $λ$, the general reduced second Zagreb index is defined by the formula $$GRM_λ(Γ)=\sum_{ab\in E(Γ)}[(°_Γ(a)+λ)(°_Γ(b)+λ)]\,.$$ A sharp lower bound for $GRM_λ$ over all trees of given order and maximum degree under the condition that $λ\ge -\frac{1}{2}$ is established. A parallel result is proved for unicyclic graphs under the condition $λ\ge -\frac{1}{2}$. The corresponding minimal trees and unicyclic graphs are identified. These findings improve upon the lower bounds previously established by Buyantogtokh, Horoldagva, and Das concerning $GRM_λ$ of trees and unicyclic graphs of given order. |
| title | Improved Lower Bounds on the General Reduced Second Zagreb Index of Trees and Unicyclic Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.30402 |