Generalized spectral Turán problems for disjoint cliques
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913045154889728 |
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| author | Xu, Yi Fan, Yi-Zheng |
| author_facet | Xu, Yi Fan, Yi-Zheng |
| contents | The generalized Turán number $\text{ex}(n, H, F)$ denotes the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Let $kK_{r+1}$ be the disjoint union of $k$ copies of the complete graph $K_{r+1}$. Recently, Gerbner determined $\text{ex}(n, K_{t},kK_{r+1})$ for all sufficiently large $n$. In this paper, we study a spectral analogue of this problem via the $t$-clique tensor of a graph. We prove that if an $n$-vertex $kK_{r+1}$-free graph $G$ maximizes the $t$-clique spectral radius, then for sufficiently large $n$, $G$ is the join of a complete graph $K_{k-1}$ and the $r$-partite Turán graph $T_{r}(n-k+1)$. This establishes a spectral counterpart of Gerbner's Theorem. Moreover, in the case $t=2$, our result recovers a theorem of Ni, Wang, and Kang on the maximum spectral radius of $kK_{r+1}$-free graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_17242 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generalized spectral Turán problems for disjoint cliques Xu, Yi Fan, Yi-Zheng Combinatorics 05C35, 05C50, 15A69 The generalized Turán number $\text{ex}(n, H, F)$ denotes the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Let $kK_{r+1}$ be the disjoint union of $k$ copies of the complete graph $K_{r+1}$. Recently, Gerbner determined $\text{ex}(n, K_{t},kK_{r+1})$ for all sufficiently large $n$. In this paper, we study a spectral analogue of this problem via the $t$-clique tensor of a graph. We prove that if an $n$-vertex $kK_{r+1}$-free graph $G$ maximizes the $t$-clique spectral radius, then for sufficiently large $n$, $G$ is the join of a complete graph $K_{k-1}$ and the $r$-partite Turán graph $T_{r}(n-k+1)$. This establishes a spectral counterpart of Gerbner's Theorem. Moreover, in the case $t=2$, our result recovers a theorem of Ni, Wang, and Kang on the maximum spectral radius of $kK_{r+1}$-free graphs. |
| title | Generalized spectral Turán problems for disjoint cliques |
| topic | Combinatorics 05C35, 05C50, 15A69 |
| url | https://arxiv.org/abs/2604.17242 |