Extremal spectral results of planar graphs without $C_{l, l}$ or $\mathrm{Theta}$ graph
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910411713937408 |
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| author | Zhang, HaoRan Wang, WenHuan |
| author_facet | Zhang, HaoRan Wang, WenHuan |
| contents | Let $\mathcal {F}$ be a given family of graphs. A graph $G$ is $\mathcal {F}$-free if it does not contain any member of $\mathcal {F}$ as a subgraph. Let $C_{l, l}$ be a graph obtained from $2C_l$ such that the two cycles share a common vertex, where $l\geqslant3 $. A $\mathrm{Theta}$ graph is obtained from a cycle $C_k$ by adding an additional edge between two non-consecutive vertices on $C_k$, where $k\geqslant 4$. Let $Θ_k$ be the set of $\mathrm{Theta}$ graphs on $k$ vertices, where $k \geqslant 4$. For sufficiently large $n $, the unique extremal planar graph with the maximum spectral radius among $C_{l, l}$-free planar graphs on $n$ vertices and among $Θ_k$-free planar graphs on $n$ vertices are characterized respectively, where $l \geqslant 3$ and $k \geqslant 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_10163 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extremal spectral results of planar graphs without $C_{l, l}$ or $\mathrm{Theta}$ graph Zhang, HaoRan Wang, WenHuan Combinatorics Let $\mathcal {F}$ be a given family of graphs. A graph $G$ is $\mathcal {F}$-free if it does not contain any member of $\mathcal {F}$ as a subgraph. Let $C_{l, l}$ be a graph obtained from $2C_l$ such that the two cycles share a common vertex, where $l\geqslant3 $. A $\mathrm{Theta}$ graph is obtained from a cycle $C_k$ by adding an additional edge between two non-consecutive vertices on $C_k$, where $k\geqslant 4$. Let $Θ_k$ be the set of $\mathrm{Theta}$ graphs on $k$ vertices, where $k \geqslant 4$. For sufficiently large $n $, the unique extremal planar graph with the maximum spectral radius among $C_{l, l}$-free planar graphs on $n$ vertices and among $Θ_k$-free planar graphs on $n$ vertices are characterized respectively, where $l \geqslant 3$ and $k \geqslant 4$. |
| title | Extremal spectral results of planar graphs without $C_{l, l}$ or $\mathrm{Theta}$ graph |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.10163 |