Maximum spread of $K_{s,t}$-minor-free graphs
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916551626588160 |
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| author | Linz, William Lu, Linyuan Wang, Zhiyu |
| author_facet | Linz, William Lu, Linyuan Wang, Zhiyu |
| contents | The spread of a graph $G$ is the difference between the largest and smallest eigenvalue of the adjacency matrix of $G$. In this paper, we consider the family of graphs which contain no $K_{s,t}$-minor. We show that for any $t\geq s \geq 2$ and sufficiently large $n$, there is an integer $ξ_{t}$ such that the extremal $n$-vertex $K_{s,t}$-minor-free graph attaining the maximum spread is the graph obtained by joining a graph $L$ on $(s-1)$ vertices to the disjoint union of $\lfloor \frac{2n+ξ_{t}}{3t}\rfloor$ copies of $K_t$ and $n-s+1 - t\lfloor \frac{2n+ξ_t}{3t}\rfloor$ isolated vertices. Furthermore, we give an explicit formula for $ξ_{t}$ and an explicit description for the graph $L$ for $t \geq \frac32(s-3) +\frac{4}{s-1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18969 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximum spread of $K_{s,t}$-minor-free graphs Linz, William Lu, Linyuan Wang, Zhiyu Combinatorics The spread of a graph $G$ is the difference between the largest and smallest eigenvalue of the adjacency matrix of $G$. In this paper, we consider the family of graphs which contain no $K_{s,t}$-minor. We show that for any $t\geq s \geq 2$ and sufficiently large $n$, there is an integer $ξ_{t}$ such that the extremal $n$-vertex $K_{s,t}$-minor-free graph attaining the maximum spread is the graph obtained by joining a graph $L$ on $(s-1)$ vertices to the disjoint union of $\lfloor \frac{2n+ξ_{t}}{3t}\rfloor$ copies of $K_t$ and $n-s+1 - t\lfloor \frac{2n+ξ_t}{3t}\rfloor$ isolated vertices. Furthermore, we give an explicit formula for $ξ_{t}$ and an explicit description for the graph $L$ for $t \geq \frac32(s-3) +\frac{4}{s-1}$. |
| title | Maximum spread of $K_{s,t}$-minor-free graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2404.18969 |