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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.21791 |
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| _version_ | 1866917294082359296 |
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| author | Ma, Mingyuan Ren, Han |
| author_facet | Ma, Mingyuan Ren, Han |
| contents | Let $G$ be a connected graph. Let $N(G)$ and $S(G)$ be the number of connected sets of $G$ and the sum of the orders of these connected sets of $G$, respectively. Then $A(G)=\frac{S(G)}{N(G)}$ is called the average order of a connected set of $G$. In this paper, we derive a closed-form formula for $A(K_m \times P_n)$, where $K_m \times P_n$ is the Cartesian product of the complete graph $K_m$ and the path $P_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21791 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The average order of a connected vertex set in $K_m \times P_n$ Ma, Mingyuan Ren, Han Combinatorics Let $G$ be a connected graph. Let $N(G)$ and $S(G)$ be the number of connected sets of $G$ and the sum of the orders of these connected sets of $G$, respectively. Then $A(G)=\frac{S(G)}{N(G)}$ is called the average order of a connected set of $G$. In this paper, we derive a closed-form formula for $A(K_m \times P_n)$, where $K_m \times P_n$ is the Cartesian product of the complete graph $K_m$ and the path $P_n$. |
| title | The average order of a connected vertex set in $K_m \times P_n$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2602.21791 |