Mutual visibility in Moore graphs and $(d,2)$-graphs with defect
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909005750730752 |
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| author | B, Tonny K M, Shikhi |
| author_facet | B, Tonny K M, Shikhi |
| contents | The concept of mutual visibility in a graph encodes combinatorial information about vertex subsets with prescribed visibility properties and serves as a useful algebraic invariant. In this paper, we derive algebraic conditions for the mutual-visibility number of $(d,2)$-graphs with non-negative defect. We then determine this parameter for $(d,2,-2)$-graphs for $d=3$ and $4$, and establish an upper bound for $d=5$. In the case $δ=0$, that is, for Moore graphs of diameter $2$, we focus on the Hoffman-Singleton graph. We establish an upper bound of $20$ for its mutual-visibility number and subsequently employ an integer programming approach to show that this bound is tight. As a corollary, we deduce that the maximum size of an induced matching in the Hoffman--Singleton graph is $10$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25858 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mutual visibility in Moore graphs and $(d,2)$-graphs with defect B, Tonny K M, Shikhi Combinatorics 05C30, 05C35 The concept of mutual visibility in a graph encodes combinatorial information about vertex subsets with prescribed visibility properties and serves as a useful algebraic invariant. In this paper, we derive algebraic conditions for the mutual-visibility number of $(d,2)$-graphs with non-negative defect. We then determine this parameter for $(d,2,-2)$-graphs for $d=3$ and $4$, and establish an upper bound for $d=5$. In the case $δ=0$, that is, for Moore graphs of diameter $2$, we focus on the Hoffman-Singleton graph. We establish an upper bound of $20$ for its mutual-visibility number and subsequently employ an integer programming approach to show that this bound is tight. As a corollary, we deduce that the maximum size of an induced matching in the Hoffman--Singleton graph is $10$. |
| title | Mutual visibility in Moore graphs and $(d,2)$-graphs with defect |
| topic | Combinatorics 05C30, 05C35 |
| url | https://arxiv.org/abs/2510.25858 |