Spectral Turán problem for $\mathcal{K}_{3,3}^{-}$-free signed graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912525924171776 |
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| author | Qin, Mingsong Li, Dan |
| author_facet | Qin, Mingsong Li, Dan |
| contents | The classical spectral Turán problem is to determine the maximum spectral radius of an $\mathcal{F}$-free graph of order $n$. Zhai and Wang [Linear Algebra Appl, 437 (2012) 1641-1647] determined the maximum spectral radius of ${C}_{4}$-free graphs of given order. Additionally, Nikiforov obtained spectral strengthenings of the Kővari-Sós-Turán theorem [Linear Algebra Appl, 432 (2010) 1405-1411] when the forbidden graphs are complete bipartite. The spectral Turán problem concerning forbidden complete bipartite graphs in signed graphs has also attracted considerable attention. Let $\mathcal{K}_{s,t}^-$ be the set of all unbalanced signed graphs with underlying graphs $K_{s,t}$. Since the cases where $s=1$ or $t=1$ do not conform to the definition of $\mathcal{K}_{s,t}^-$, it follows that $s,t\geq 2$. Wang and Lin [Discrete Appl. Math, 372 (2025) 164-172] have solved the case of $s=t=2$ since $\mathcal{K}_{2,2}^-$ is $\mathcal{C}_{4}^{-}$ in this situation. This paper gives an answer for $s=t=3$ and completely characterizes the corresponding extremal signed graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_05500 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral Turán problem for $\mathcal{K}_{3,3}^{-}$-free signed graphs Qin, Mingsong Li, Dan Combinatorics The classical spectral Turán problem is to determine the maximum spectral radius of an $\mathcal{F}$-free graph of order $n$. Zhai and Wang [Linear Algebra Appl, 437 (2012) 1641-1647] determined the maximum spectral radius of ${C}_{4}$-free graphs of given order. Additionally, Nikiforov obtained spectral strengthenings of the Kővari-Sós-Turán theorem [Linear Algebra Appl, 432 (2010) 1405-1411] when the forbidden graphs are complete bipartite. The spectral Turán problem concerning forbidden complete bipartite graphs in signed graphs has also attracted considerable attention. Let $\mathcal{K}_{s,t}^-$ be the set of all unbalanced signed graphs with underlying graphs $K_{s,t}$. Since the cases where $s=1$ or $t=1$ do not conform to the definition of $\mathcal{K}_{s,t}^-$, it follows that $s,t\geq 2$. Wang and Lin [Discrete Appl. Math, 372 (2025) 164-172] have solved the case of $s=t=2$ since $\mathcal{K}_{2,2}^-$ is $\mathcal{C}_{4}^{-}$ in this situation. This paper gives an answer for $s=t=3$ and completely characterizes the corresponding extremal signed graphs. |
| title | Spectral Turán problem for $\mathcal{K}_{3,3}^{-}$-free signed graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2508.05500 |