The $α$-spectral Turán type problems for graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911296392265728 |
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| author | Wu, Jiadong Lu, Yongchun Kang, Liying |
| author_facet | Wu, Jiadong Lu, Yongchun Kang, Liying |
| contents | For $0 \leq α< 1$, the $α$-spectral radius of a graph $G$ is defined as
the largest eigenvalue of $A_α(G)=αD(G)+(1-α)A(G)$, where $D(G)$ and $A(G)$ are the diagonal matrix of degrees and adjacency matrix of $G$, respectively.
A graph is called color-critical if it contains an edge whose deletion reduces its chromatic number.
The celebrated Erdős-Stone-Simonovits theorem asserts that $ \mathrm{ex}(n,\mathcal{F})=\left(1-\frac{1}{χ(\mathcal{F})-1}+o(1)\right)\frac{n^2}{2},$
where $χ(\mathcal{F})$ is the chromatic number of $\mathcal{F}$.
Nikiforov and Zheng et al. established the adjacency spectral and signless Laplacian spectral versions of this theorem, respectively.
In this paper, we present the $α$-spectral version of this theorem, which unifies the aforementioned results. Furthermore, we characterize the $α$-spectral extremal graphs for color-critical graphs, thereby extending the existing results on adjacency spectral and signless Laplacian spectral extremal graphs for such graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_01673 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $α$-spectral Turán type problems for graphs Wu, Jiadong Lu, Yongchun Kang, Liying Combinatorics 05C50, 05C35 For $0 \leq α< 1$, the $α$-spectral radius of a graph $G$ is defined as the largest eigenvalue of $A_α(G)=αD(G)+(1-α)A(G)$, where $D(G)$ and $A(G)$ are the diagonal matrix of degrees and adjacency matrix of $G$, respectively. A graph is called color-critical if it contains an edge whose deletion reduces its chromatic number. The celebrated Erdős-Stone-Simonovits theorem asserts that $ \mathrm{ex}(n,\mathcal{F})=\left(1-\frac{1}{χ(\mathcal{F})-1}+o(1)\right)\frac{n^2}{2},$ where $χ(\mathcal{F})$ is the chromatic number of $\mathcal{F}$. Nikiforov and Zheng et al. established the adjacency spectral and signless Laplacian spectral versions of this theorem, respectively. In this paper, we present the $α$-spectral version of this theorem, which unifies the aforementioned results. Furthermore, we characterize the $α$-spectral extremal graphs for color-critical graphs, thereby extending the existing results on adjacency spectral and signless Laplacian spectral extremal graphs for such graphs. |
| title | The $α$-spectral Turán type problems for graphs |
| topic | Combinatorics 05C50, 05C35 |
| url | https://arxiv.org/abs/2512.01673 |