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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2402.15778 |
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| _version_ | 1866913249764573184 |
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| author | An, Junfeng Tian, Yingzhi |
| author_facet | An, Junfeng Tian, Yingzhi |
| contents | The Wiener index $W(G)$ of a graph $G$ is one of the most well-known topological indices, which is defined as the sum of distances between all pairs of vertices of $G$. The diameter $D(G)$ of $G$ is the maximum distance between all pairs of vertices of $G$; the conditional diameter $D(G;s)$ is the maximum distance between all pairs of vertex subsets with cardinality $s$ of $G$. When $s=1$, the conditional diameter $D(G;s)$ is just the diameter $D(G)$. The authors in \cite{QS} characterized the graphs with the maximum Wiener index among all graphs with diameter $D(G)=n-c$, where $1\le c\le 4$. In this paper, we will characterize the graphs with the maximum Wiener index among all graphs with conditional diameter $D(G;s)=n-2s-c$ ( $-1\leq c\leq 1$), which extends partial results in \cite{QS}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15778 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Graphs with a given conditional diameter that maximize the Wiener index An, Junfeng Tian, Yingzhi Combinatorics The Wiener index $W(G)$ of a graph $G$ is one of the most well-known topological indices, which is defined as the sum of distances between all pairs of vertices of $G$. The diameter $D(G)$ of $G$ is the maximum distance between all pairs of vertices of $G$; the conditional diameter $D(G;s)$ is the maximum distance between all pairs of vertex subsets with cardinality $s$ of $G$. When $s=1$, the conditional diameter $D(G;s)$ is just the diameter $D(G)$. The authors in \cite{QS} characterized the graphs with the maximum Wiener index among all graphs with diameter $D(G)=n-c$, where $1\le c\le 4$. In this paper, we will characterize the graphs with the maximum Wiener index among all graphs with conditional diameter $D(G;s)=n-2s-c$ ( $-1\leq c\leq 1$), which extends partial results in \cite{QS}. |
| title | Graphs with a given conditional diameter that maximize the Wiener index |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2402.15778 |