On the transmission irregular trees with the maximum Wiener index
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917219114418176 |
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| author | Damnjanović, Ivan Xu, Anran Xu, Kexiang |
| author_facet | Damnjanović, Ivan Xu, Anran Xu, Kexiang |
| contents | The transmission of a vertex $v$ in a (chemical) graph $G$ is the sum of distances from $v$ to other vertices in $G$. If any two vertices of $G$ have different transmissions, then $G$ is transmission irregular. The Wiener index $W(G)$ of a graph $G$ is the sum of all distances between all unordered pairs of vertices in $G$, which has another formula as the half of the sum of transmissions of all vertices of $G$. In this paper, we consider the Wiener index maximization problem on the set of transmission irregular trees of a given order $n \in \mathbb{N}$. We solve the problem for all odd values of $n$ and for almost all even values of $n$. Each resolved extremal problem has a unique solution that is a chemical tree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13585 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the transmission irregular trees with the maximum Wiener index Damnjanović, Ivan Xu, Anran Xu, Kexiang Combinatorics 05C05, 05C12, 05C35 The transmission of a vertex $v$ in a (chemical) graph $G$ is the sum of distances from $v$ to other vertices in $G$. If any two vertices of $G$ have different transmissions, then $G$ is transmission irregular. The Wiener index $W(G)$ of a graph $G$ is the sum of all distances between all unordered pairs of vertices in $G$, which has another formula as the half of the sum of transmissions of all vertices of $G$. In this paper, we consider the Wiener index maximization problem on the set of transmission irregular trees of a given order $n \in \mathbb{N}$. We solve the problem for all odd values of $n$ and for almost all even values of $n$. Each resolved extremal problem has a unique solution that is a chemical tree. |
| title | On the transmission irregular trees with the maximum Wiener index |
| topic | Combinatorics 05C05, 05C12, 05C35 |
| url | https://arxiv.org/abs/2512.13585 |