The spectral Turán problem: Characterizing spectral-consistent graphs
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
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2025
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| _version_ | 1866917355836145664 |
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| author | Fang, Longfei Goryainov, Sergey Krotov, Denis Lin, Huiqiu Zhai, Mingqing |
| author_facet | Fang, Longfei Goryainov, Sergey Krotov, Denis Lin, Huiqiu Zhai, Mingqing |
| contents | Let ${\rm EX}(n,H)$ and ${\rm SPEX}(n,H)$ denote the families of $n$-vertex $H$-free graphs with the maximum size and the maximum spectral radius, respectively. A graph $H$ is said to be spectral-consistent if ${\rm SPEX}(n,H)\subseteq {\rm EX}(n,H)$ for sufficiently large $n$. A fundamental problem in spectral extremal graph theory is to determine which graphs are spectral-consistent. Cioabă, Desai and Tait [European J. Combin. 99 (2022) 103420] proposed the following conjecture: Let $H$ be any graph such that the graphs in ${\rm EX}(n,H)$ are Turán graph plus $O(1)$ edges. Then $H$ is spectral-consistent. Wang, Kang and Xue [J. Combin. Theory Ser. B 159 (2023) 20--41] confirmed this conjecture, along with a stronger result. Recently, Liu and Ning raised a general problem in spectral extremal graph theory: Characterize all graphs that are spectral-consistent.
In this paper, we establish that for any finite graph \(H\), if its decomposition family is matching-good, then \(H\) is necessarily spectral-consistent. Notably, this structural condition is strictly weaker than the condition for spectral-consistency established by Wang, Kang, and Xue in their earlier work, thereby broadening the class of graphs known to satisfy the spectral-consistency property. Our main result enables us to fully characterize the spectral-consistency for several important families of forbidden graphs \(H\), including generalized color-critical graphs, odd-ballooning of trees and complete bipartite graphs, as well as edge blow-up of non-bipartite graphs and certain special bipartite graphs. Furthermore, we present a streamlined proof for an existing spectral-consistency result due to Chen, Lei, and Li, simplifying their original argument. Finally, we propose several open problems to motivate future research in this area. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12070 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The spectral Turán problem: Characterizing spectral-consistent graphs Fang, Longfei Goryainov, Sergey Krotov, Denis Lin, Huiqiu Zhai, Mingqing Combinatorics Let ${\rm EX}(n,H)$ and ${\rm SPEX}(n,H)$ denote the families of $n$-vertex $H$-free graphs with the maximum size and the maximum spectral radius, respectively. A graph $H$ is said to be spectral-consistent if ${\rm SPEX}(n,H)\subseteq {\rm EX}(n,H)$ for sufficiently large $n$. A fundamental problem in spectral extremal graph theory is to determine which graphs are spectral-consistent. Cioabă, Desai and Tait [European J. Combin. 99 (2022) 103420] proposed the following conjecture: Let $H$ be any graph such that the graphs in ${\rm EX}(n,H)$ are Turán graph plus $O(1)$ edges. Then $H$ is spectral-consistent. Wang, Kang and Xue [J. Combin. Theory Ser. B 159 (2023) 20--41] confirmed this conjecture, along with a stronger result. Recently, Liu and Ning raised a general problem in spectral extremal graph theory: Characterize all graphs that are spectral-consistent. In this paper, we establish that for any finite graph \(H\), if its decomposition family is matching-good, then \(H\) is necessarily spectral-consistent. Notably, this structural condition is strictly weaker than the condition for spectral-consistency established by Wang, Kang, and Xue in their earlier work, thereby broadening the class of graphs known to satisfy the spectral-consistency property. Our main result enables us to fully characterize the spectral-consistency for several important families of forbidden graphs \(H\), including generalized color-critical graphs, odd-ballooning of trees and complete bipartite graphs, as well as edge blow-up of non-bipartite graphs and certain special bipartite graphs. Furthermore, we present a streamlined proof for an existing spectral-consistency result due to Chen, Lei, and Li, simplifying their original argument. Finally, we propose several open problems to motivate future research in this area. |
| title | The spectral Turán problem: Characterizing spectral-consistent graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2508.12070 |